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GENERAL 
PHYSICS 

Mechanics and 
Molecular Physics 



Pergamon J 




General Physics 

Mechanics and Molecular Physics 



L.D, Landau, A.L Akhiezer and 
EM. Lifshitz 



This is a general physics textbook written 
at first and second year undergraduate 
level, covering mechanics, symmetry 
theory, heat and thermodynamics, and 
solid-state physics. The subjects are 
covered under the following headings: 
Particle Mechanics ; Fields ; Motion of a 
Rigid Body ; Oscillations ; The Structure 
of M atter ; The Theory of Symmetry ; 
H eat ; Thermal Processes ; Phase Transi - 
tions ; Solutions ; Chemical Reactions ; 
Surface Phenomena ; Mechanical 
Properties of Solids ; Diffusion and 
Thermal Conduction ; Viscosity. 

The book emphasizes the close relation- 
ship of physics to physical chemistry, 
crystallography and the properties of 
matter to a greater extent than is usual in 
textbooks of this standard. 
The approach is also different, and the 
treatment is distinguished by the unique 
insight and illumination characteristic of 
Landau and his school, and already well 
known to students of physics through the 
volumes in Landau and Lifshitz's Course 
of Theoretical Physics, published by 
Pergamon Press. 





L, D. LANDAU, A. I. AKHIEZER 
E. M. LIFSHITZ 



GENERAL PHYSICS 

Mechanics and Molecular Physics 

Translated by 

J. B. SYKES 

A. D, PETFORD, C. L. PETFORD 



PERGAMON PRESS 

OXFORD • LONDON ' EDINBURGH -NEW YORK 
TORONTO • SYDNEY - PARIS ■ BRAUNSCHWEIG 



Pergamon Press Ltd.. Headington Hill Hall, Oxford 
4 & 5 Fitzroy Square, London W, I 

Pergamon Press (Scotland) Ltd., 2 & 3 Tevjot Place. Edinburgh I 

Pergamon Press Inc., 44-01 2 1st Street, Long Island City, New York 11 101 

Pergamon of Canada, Ltd.. 6 Adelaide Street East, Toronto, Ontario 

Pergamon Press (Anal) Pty. Ltd,, RushcuUers Bay, Sydney, N.S.W. 

Pergamon Press S. A.R.L,, 24 rue des Ecoles, Paris 5* 

Vieweg & Sohn GmbH, Burgplaiz 1 , Braunschweig 



Copyright C 1967 
Pergamon Press Ltd. 



First English edition 1967 



Tr&fifllatod From Kurs obshcheifiziki; meklmnikti i nwtekulyarnaya fiziku, 
IzdatePslvo "Nauka", Moscow, 1965 



Library of Congress Catalog No. 67-30260 



Filmsct by Graphic Film Limited, Dublin 

and printed in Great Britain by 

J. W. Arrowsmith Limited, Bristol 



IW 003 304 o 



CONTENTS 



Preface 



ix 



t PARTICLE MECHANICS 

§1. The principle of the relativity of motion 1 

§2. Velocity 3 

§3. Momentum 5 

$4. Motion under reactive forces 7 

§5. Centre of mass g 

§6. Acceleration 10 

§7. Force 12 

§8. Dimensions of physical quantities 15 

$9. Motion in a uniform field 19 

§10. Work and potential energy 20 

§11. The law of conservation of energy 23 

§12. Internal energy 26 

§13. Boundaries of the motion 27 

§14. Elastic collisions 31 

§15. Angular momentum 36 

§16. Motion in a central field 40 

II. FIELDS 

§17. Electrical interaction 44 

§18. Electric field 46 

§19. Electrostatic potential 49 

§20. Gauss' theorem 5] 

§21. Electric fields in simple cases 53 

§22. Gravitational field 56 

§23. The principle of equivalence 60 

§24. Kepleri an motion 62 

III. MOTION Oh A RIGID BODY 

§25. Types of motion of a rigid body 66 

§26, The energy of a rigid body in motion 68 

§27. Rotational angular momentum 72 

§28. The equation of motion of a rotating body 73 

§29. Resultant force 77 

§30. The gyroscope 78 

§31. Inertia forces g| 

IV. OSCILLATIONS 

§32, Simple harmonic oscillations 86 

§33. The pendulum 90 

§34, Damped oscillations 93 



VI CONTENTS 

§35. Forced oscillations 96 

§36, Parame tri c res onance 1 02 

V, THE STRUCTURE OF MATTER 

§37. Atoms 105 

§38. Isotopes 1° 9 

§39. Molecules ' ' l 

VI. THE THEORY OF SYMMETRY 

§40. Symmetry of molecules I 1 5 

§41. Stereoisomerism I IK 

§42. Crystal lattices 120 

§43. Crystal systems 123 

§44. Space groups 129 

§45. Crystal classes 131 

§46. Lattices of the chemical elements 133 

§47. Lattices of compounds 137 

§48. Crystal planes 139 

§49. The natural boundary of a crystal 142 

VII. HEAT 

§50. Temperature 144 

§51. Pressure 149 

§52. Aggregate states of matter 151 

§53. Ideal gases 153 

§54. An ideal gas in an external field 157 

§55. The Maxwellian distribution 160 

§56. Work and quantity of heal 166 

§57. The specific heat of gases 171 

§58. Solids and liquids 174 

VIII. THERMAL PROCESSES 

§59. A diabatic processes 178 

§60. Joule- Kelvin processes 182 

§61. Steady flow 184 

§62. Irreversibility of thermal processes 187 

§63. The Carnot cycle 190 

§64. The nature of irreversibility 192 

§65. Entropy '94 

IX. PHASE TRANSITIONS 

§66. Phases of matter 197 

§67. The Clausius-Clapeyron equation 201 

§68. Evaporation 203 

§69. The critical point 207 

§70. Van derWaals' equation 210 

§71. The law of corresponding states 214 

§72. The triple point 216 

£73. Crystal modifications . 218 

§74. Phase transitions of the second kind 222 

575. Ordering of crystals 225 

§76. Liquid crystals 227 



CONTENTS Vii 

X. SOLUTIONS 

§77. Solubility 230 

§78. Mixtures of liquids 232 

§79, Solid solutions 234 

§80, Osmotic pressure 236 

§8). Raouli'slaw 238 

$82. Boding of a mixture of liquids 241 

$83, Reverse condensation 244 

$84. Solidification of a mixture of liquids 246 

$85. The phase rule 250 

XI. CHEMICAL REACTIONS 

486. Heats of reaction 252 

$87. Chemical equilibrium 254 

$88. The law of mass action 256 

$89. Strong electrolytes 262 

§90. Weak electrolytes 264 

§91. Activation energy 266 

§92. Molecularity of reactions 270 

§93. Chain reactions 272 

XII. SURFACE PHENOMENA 

$94. Surface tension 276 

$95. Adsorption 279 

$96. Angle of contact 282 

$97. Capillary forces 285 

§98. Vapour pressure over a curved surface 288 

§ 99 . The n atu re of su perhealing and su pe rcool i ng 289 

$100. Colloidal solutions 291 

XIII. MECHANICAL PROPERTIES OF SOLIDS 

$101. Extension 294 

§102. Uniform compression 298 

§103. Shear 301 

$104. Plasticity 305 

$105. Defects in crystals 308 

$106. The nature of plasticity 312 

$107. Friction of solids 316 

XIV. DIFFUSION AND THERMAL CONDUCTION 

$108. The diffusion coefficient 318 

$109. The thermal conductivity 319 

^ 1 10. Thermal resistance 321 

§111. The equalisation time 326 

§112. The mean free path 328 

$ 1 1 3. Diffusion and thermal conduction in gases 330 

$114, Mobility 334 

$115. Thermal diffusion 336 

$Lt6. Diffusion in solids 338 

XV. VISCOSITY 

§117. The coefficient of viscosity 341 

§118. Viscosity of gases and liquids 343 



vni 



CONTENTS 



§ 1 1 9. Poiseuilte's formula 

§ 1 20. The similarity method *** 

§121. Siokes' formula JJJ 

§122, Turbulence "* 

§123. Rarefied gases ^' 

§124. Superfluidity 3bl 

Index 367 



PREFACE 



The purpose of this book is to acquaint the reader with the 
principal phenomena and most important laws of physics. The 
authors have tried to make the book as compact as possible, 
including only what is essential and omitting what is of secondary 
significance. For this reason the discussion nowhere aims at 
anything approaching completeness. 

The derivations of ihe formulae are given only in so far as they 
may help the reader in undemanding the relations between 
phenomena. Formulae are therefore derived for simple cases 
wherever possible, on the principle that the systematic derivation 
of quantitative formulae and equations should rather appear in a 
textbook of theoretical physics. 

The reader is assumed to be familiar with algebra and trig- 
onometry and also to understand the fundamentals of the 
differential calculus and of vector algebra. He is further expected 
to have an initial knowledge of the main ideas of physics and 
chemistry. The authors hope that the book will be useful to physics 
students at universities and technical colleges, and also to physics 
teachers in schools. 

This book was originally written in 1937, but has not been 
published until now, for a variety of reasons. It has now been 
augmented and rewritten, but the plan and essential content 
remain unchanged. 

To our profound regret, L. D. Landau, our teacher and friend, 
has been prevented by injuries received in a road accident from 
personally contributing to the preparation of this edition. We have 
everywhere striven to follow the manner of exposition that is 
characteristic of him. 

We have also attempted to retain as far as possible the original 
choice of material, being guided here both by the book in its 
original form and by the notes (published in 1948) taken from 
Landau's lectures on general physics in the Applied Physics 
Department of Moscow State University. 

ix 



X PREFACE 

In the original plan, in order not to interrupt the continuity of 
the discussion, the methods of experimental study of thermal 
phenomena were to have been placed in a separate chapter at the 
end of the book. Unfortunately, we have not yet had an oppor- 
tunity to carry out this intention, and we have decided, in order 
to avoid further delay, to publish the book without that chapter. 

A. I. Akhiezer 
June 1965 E. M. Lifshitz 



CHAPTER I 

PARTICLE MECHANICS 



§ 1 . The principle of the relativity of motion 

The fundamental concept of mechanics is that of motion of a 
body with respect to other bodies. In the absence of such other 
bodies it is clearly impossible to speak of motion, which is always 
relative. Absolute motion of a body irrespective of other bodies 
has no meaning. 

The relativity of motion arises from the relativity of the concept 
of space itself. We cannot speak of position in absolute space 
independently of bodies therein, but only of position relative to 
certain bodies. 

A group of bodies which are arbitrarily considered to be at 
rest, the motion of other bodies being taken as relative to that 
group, is called in physics a frame of reference. A frame of 
reference may be arbitrarily chosen in an infinite number of 
ways, and the motion of a given body in different frames will in 
general be different. If the frame of reference is the body itself, 
then the body will be at rest in that frame, while in other frames 
it will be in motion, and in different frames it will move differently, 
i.e. along different paths. 

Different frames of reference are equally valid and equally 
admissible for investigating the motion of any given body. 
Physical phenomena, however, in general occur differently in 
different frames, and in this way different frames of reference may 
be distinguished. 

It is reasonable to choose the frame of reference such that 
natural phenomena take their simplest form. Let us consider 
a body so far from other bodies that it does not interact with 
them. Such a body is said to be moving freely. 

In reality, the condition of free motion can, of course, be 
fulfilled only to a certain approximation, but we can imagine in 



2 PARTICLE MECHANICS [i 

principle that a body is free from interaction with other bodies to 
any desired degree of accuracy. 

Free motion, like other forms of motion, appears differently in 
different frames of reference. If, however, the frame of reference 
is one in which any one freely moving body is fixed, then free 
motion of other bodies is especially simple: it is uniform motion 
in a straight line or, as it is sometimes called, motion with a 
velocity constant in magnitude and direction. This statement 
forms the content of the law of inertia, first stated by Galileo. A 
frame of reference in which a freely moving body is fixed is called 
an inertial frame. The law of inertia is also known as Newton's 
first law. 

It might appear at first sight that the definition of an inertial 
frame as one with exceptional properties would permit a defini- 
tion of absolute space and absolute rest relative to that frame. 
This is not so, in fact, since there exists an infinity of inertial 
frames: if a frame of reference moves with a velocity constant 
in magnitude and direction relative to an inertial frame, then it is 
itself an inertial frame. 

It must be emphasised that the existence of inertial frames of 
reference is not purely a logical necessity. The assertion that 
there exist, in principle, frames of reference with respect to which 
the free motion of bodies takes place uniformly and in a straight 
line is one of the fundamental laws of Nature. 

By considering free motion we evidently cannot distinguish 
between different inertial frames. It may be asked whether the 
examination of other physical phenomena might in some way 
distinguish one inertial frame from another and hence select 
one frame as having special properties. If this were possible, we 
could say that there is absolute space and absolute rest relative 
to this special frame of reference. There is, however, no such 
distinctive frame, since all physical phenomena occur in the same 
way in different inertial frames. 

All the laws of Nature have the same form in every inertial 
frame, which is therefore physically indistinguishable from, and 
equivalent to, every other inertial frame. This result, one of the 
most important in physics, is called the principle of relativity of 
motion, and deprives of all significance the concepts of absolute 
space, absolute rest and absolute motion. 

Since all physical laws are formulated in the same way in 



§2] VELOCITY 3 

every inertial frame, but in different ways in different non-inertial 
frames, it is reasonable to study any physical phenomenon in 
inertial frames, and we shall do so henceforward except where 
otherwise stated. 

The frames of reference actually used in physical experiments 
are inertial only to a certain approximation. For example, the 
most usual frame of reference is that in which the Earth, on which 
we live, is fixed. This frame is not inertial, owing to the daily 
rotation of the Earth on its axis and its revolution round the 
Sun. These motions occur with different and varying velocities 
at different points on the Earth, and the frame in which the Earth 
is fixed is therefore not inertial. However, because of the relative 
slowness of variation of the direction of the velocities in the 
Earth's daily rotation on its axis and revolution round the Sun, 
we in fact commit a very small error, of no importance in many 
physical experiments, by assuming that the "terrestrial" frame of 
reference is an inertial frame. Although the difference between 
the motion in the terrestrial frame of reference and that in an 
inertial frame is very slight, it can nevertheless be observed, for 
example, by means of a Foucault pendulum, whose plane of 
oscillation slowly moves relative to the Earth's surface (§31). 

§2. Velocity 

It is reasonable to begin the study of the laws of motion by 
considering the motion of a body of small dimensions. The 
motion of such a body is especially simple because there is no 
need to take into account the rotation of the body or the relative 
movement of different parts of the body. 

A body whose size may be neglected in considering its motion 
is called a particle, and is a fundamental object of study in 
mechanics. The possibility of treating the motion of a given body 
as that of a particle depends not only on its absolute size but also 
on the conditions of the physical problem concerned. For 
example, the Earth may be regarded as a particle in relation to 
its motion round the Sun, but not in relation to its daily rotation 
on its axis. 

The position of a particle in space is entirely defined by 
specifying three coordinates, for instance the three Cartesian 
coordinates x, y, z. For this reason a particle is said to have three 
degrees of freedom. The quantities x, y, z form the radius vector 



4 PARTICLE MECHANICS |I 

r from the origin to the position of the particle. 

The motion of a particle is described by its velocity. In uniform 
motion, the velocity is denned simply as the distance traversed 
by the particle in unit time. Generally, when the motion is not 
uniform and varies in direction, the particle velocity must be 
defined as a vector equal to the vector of an infinitesimal displace- 
ment ds of the particle divided by the corresponding infinitesimal 
time interval dt. Denoting the velocity vector by v, we therefore 
have 

v = ds/dt. 

The direction of the vector v is the same as that of ds; that is, 
the velocity at any instant is along the tangent to the path of the 
particle in the direction of motion. 




Figure 1 shows the path of a particle and the radius vectors 
r and r + dv at times t and t+dt. By the vector addition rule it is 
easily seen that the infinitesimal displacement ds of the particle 
is equal to the difference between the radius vectors at the initial 
and final instants: ds = dr. The velocity v may therefore be 
written 

v = dr/dt, 

and is thus the time derivative of the radius vector of the moving 
particle. Since the components of the radius vector r are the 
coordinates x, y, z, the components of the velocity along these 
axes are the derivatives 

v x = dx/dt, v y = dy/dt, v z = dzldt . 

The velocity, like the position, is a fundamental quantity 



§3] MOMENTUM 5 

describing the state of motion of a particle. The state of the par- 
ticle is therefore defined by six quantities: three coordinates and 
three velocity components. 

The relation between the velocities v and v' of the same particle 
in two different frames of reference K and K' may be determined 
as follows. If in a time dt the particle moves an amount ds relative 
to the frame K, and the frame K moves an amount dS relative to 
the frame K', the vector addition rule shows that the displace- 
ment of the particle relative to the frame K' is ds' = ds + dS. 
Dividing both sides by the time interval dt and denoting the 
velocity of the frame K' relative to K by V, we find 

v' = v + V. 

This formula relating the velocities of a given particle in different 
frames of reference is called the velocity addition rule. 

At first sight the velocity addition rule appears obvious, but 
in fact it depends on the tacitly made assumption that the passage 
of time is absolute. We have assumed that the time interval during 
which the particle moves by an amount ds in the frame K is equal 
to the time interval during which it moves by ds' inK'. In reality, 
this assumption proves to be not strictly correct, but the conse- 
quences of the non-absoluteness of time begin to appear only at 
very high velocities, comparable with that of light. In particular, 
the velocity addition rule is not obeyed at such velocities. In 
what follows we shall consider only velocities so small that the 
assumption of absolute time is quite justified. 

The mechanics based on the assumption that time is absolute 
is called Newtonian or classical mechanics, and we shall here 
discuss only this mechanics. Its fundamental laws were stated 
by Newton in his Principia (1687). 

§3. Momentum 

In free motion of a particle, i.e. when it does not interact with 
other bodies, its velocity remains constant in any inertial frame 
of reference. If particles interact with one another, however, their 
velocities will vary with time; but the changes in the velocities 
of interacting particles are not completely independent of one 
another. In order to ascertain the nature of the relation between 
them, we define a closed system — a. group of particles which 



6 PARTICLE MECHANICS L 1 

interact with one another but not with surrounding bodies. For a 
closed system there exist a number of quantities related to the 
velocities which do not vary with time. These quantities naturally 
play a particularly important part in mechanics. 

One of these invariant or conserved quantities is called the total 
momentum of the system. It is the vector sum of the momenta 
of each of the particles forming a closed system. The momentum 
of a single particle is simply proportional to its velocity. The 
proportionality coefficient is a constant for any given particle and 
is called its mass. Denoting the particle momentum vector by 
p and the mass by m, we can write 

p = m\, 

where v is the velocity of the particle. The sum of the vectors 
p over all particles in the closed system is the total momentum 
of the system: 

P= Pi + p 2 +- ' ' =ra 1 v 1 +ra 2 v 2 +- • •, 

where the suffixes label the individual particles and the sum 
contains as many terms as there are particles in the system. This 
quantity is constant in time: 

P = constant. 

Thus the total momentum of a closed system is conserved. 
This is the law of conservation of momentum. Since the momen- 
tum is a vector, the law of conservation of momentum separates 
into three laws expressing the constancy in time of the three 
components of the total momentum. 

The law of conservation of momentum involves a new quantity, 
the mass of a particle. By means of this law, we can determine 
the ratios of particle masses. For let us imagine a collision 
between two particles of masses m t and ra 2 , and let v x and v 2 
denote their velocities before the collision, v/ and v 2 ' their 
velocities after the collision. Then the law of conservation of 
momentum shows that 

ra^ +m 2 v 2 = m^Vi + ra 2 v 2 '. 



§4] MOTION UNDER REACTIVE FORCES 7 

If Av x and Av 2 are the changes in the velocities of the two 
particles, this relation may be written as 

AWiAvx + m 2 Av 2 = 0, 
whence 

Av 2 = — (m 1 /m 2 )Av 1 . 

Thus the changes in velocity of two interacting particles are 
inversely proportional to their masses. Using this relation, we 
can find the ratio of the masses of the particles from the changes 
in their velocities. We must therefore arbitrarily take the mass of 
some particular body as unity and express the masses of all other 
bodies in terms of it. This unit of mass in physics is usually taken 
to be the gram (see §8). 

§4. Motion under reactive forces 

The law of conservation of momentum is one of the funda- 
mental laws of Nature and plays a part in many phenomena. In 
particular, it accounts for motion under reactive forces. 

We shall show how the velocity of a rocket may be found as a 
function of its varying mass. Let the velocity of the rocket relative 
to the Earth at some instant t be v and its mass M. At this instant, 
let the rocket begin to emit exhaust gases whose velocity relative 
to the rocket is u. In a time dt the mass of the rocket decreases 
to M + dM, where — dM is the mass of the gas emitted, and the 
velocity increases to v + dv. Now let us equate the momenta of 
the system consisting of the rocket and the exhaust gases at 
times / and t+dt. The initial momentum is evidently Mv. The 




y////////////////////////////////, 

Fig. 2. 



8 PARTICLE MECHANICS [i 

momentum of the rocket at time t + dt is (M+dM)(v + dv) 
(dM being negative) and the momentum of the exhaust gas is 
—dM{v — u), since the velocity of the gas relative to the Earth 
is clearly v — u (Fig. 2). The momenta at these two times must be 
equal, by the law of conservation of momentum: 

Mv = (M + dM){v + dv) - dM(v - u), 

whence, neglecting the second-order small quantity dMdv, we 
obtain 

Mdv + udM=0 

or 

dMlM = -dvlu. 

We shall suppose that the gas outflow velocity does not vary 
with time. Then the last equation may be written 

d loggM = — d(v/u), 
and therefore 

log e M + vlu = constant. 

The value of the constant is given by the condition that the mass 
of the rocket is M at the beginning of its motion (i.e. when v = 0), 
so that 

log e M+ vlu = log e M , 

whence we have finally 

v = u log e (M /M). 

This formula gives the velocity of the rocket as a function of its 
varying mass. 

§5. Centre of mass 

The law of conservation of momentum is related to an impor- 
tant property of mass, the law of conservation of mass. In order 
to understand the meaning of this law, let us consider the point 



§5] CENTRE OF MASS 9 

called the centre of mass of a closed system of particles. The 
coordinates of the centre of mass are the mean values of the 
coordinates of the particles, the coordinate of each particle being 
counted as many times as its mass exceeds the unit mass. That 
is, if x u x 2 ,..., denote the x coordinates of particles having 
masses m u m 2 , . . . , then the jc coordinate of the centre of mass is 
determined by the formula 

_ m 1 x 2 +m 2 x 2 +- • • 



m 1 +m 2 + 

Similar formulae may be written for the y and z coordinates. 
These formulae can be put in a single vector form as an expres- 
sion for the radius vector R of the centre of mass: 

_ m 1 r 1 +m 2 r 2 +- • • 
m 1 +m 2 + 

where rj, r 2 , . . . are the radius vectors of the individual particles. 
The centre of mass has the noteworthy property of moving 
with constant velocity, whereas the individual particles forming 
the closed system may move with velocities which vary with 
time. For the velocity of the centre of mass is 

v _ dR _ m 1 dr 1 ldt+m 2 dr 2 /dt+ • • • 



dt m 1 + m 2 + • • • 

But drjdt is the velocity of the first particle, drjdt that of the 
second particle, and so on. Denoting these by \ t , v 2 , . . . , we have 

_ mxVx + m 2 v 2 -I 

m l + m 2 + - • • 

The numerator is the total momentum of the system, which we 
have denoted by P, and we therefore have finally 

V = P/M, 

where M is the total mass of the particles: M = m x + m 2 H . 

Since the total momentum of the system is conserved, the velocity 
of the centre of mass is constant in time. 



10 PARTICLE MECHANICS [i 

Writing this formula as 

P = MV, 

we see that the total momentum of the system, the velocity of its 
centre of mass and the total mass of the particles in the system 
are related in the same way as the momentum, velocity and mass 
of a single particle. We can regard the total momentum of the 
system as the momentum of a single particle at the centre of 
mass of the system, with a mass equal to the total mass of the 
particles in the system. The velocity of the centre of mass may 
be regarded as the velocity of the system of particles as a whole, 
and the sum of the individual masses appears as the mass of the 
whole system. 

Thus we see that the mass of a composite body is equal to the 
sum of the masses of its parts. This is a very familiar assertion 
and might appear to be self-evident; but it is in fact by no means 
trivial and represents a physical law which follows from the law 
of conservation of momentum. 

Since the velocity of the centre of mass of a closed system of 
particles is constant in time, a frame of reference in which the 
centre of mass is fixed is an inertial frame, called the centre-of- 
mass frame. The total momentum of a closed system of particles 
is obviously zero in this frame. The description of phenomena in 
this frame of reference eliminates complications arising from the 
motion of the system as a whole, and demonstrates more clearly 
the properties of the internal processes occurring within the 
system. For this reason the centre-of-mass frame is frequently 
used in physics. 

§6. Acceleration 

For a particle moving in a general manner the velocity varies 
continually in both magnitude and direction. Let the velocity 
change by d\ in a time dt . If the change per unit time is taken we 
have the acceleration vector of the particle, denoted here by w: 

w = d\ldt. 

Thus the acceleration determines the change in the velocity of 
the particle and is equal to the time derivative of the velocity. 



§6] 



ACCELERATION 



11 



If the direction of the velocity is constant, i.e. the particle 
moves in a straight line, then the acceleration is along that line 
and is clearly 

w = dvjdt. 

It is also easy to determine the acceleration when the velocity 
remains constant in magnitude but varies in direction. This case 
occurs when a particle moves uniformly in a circle. 




Fig. 3. 

Let the velocity of the particle at some instant be v (Fig. 3). 
We mark v from a point C on an auxiliary diagram (Fig. 4). When 
the particle moves uniformly in a circle, the end of the vector 
v (the point A) also moves uniformly in a circle of radius v equal 
to the magnitude of the velocity. It is clear that the velocity of 
A will be equal to the acceleration of the original particle P, since 
the motion of A in a time dt is d\ and its velocity is therefore dsldt. 
This velocity is tangential to the circle round C and is perpen- 
dicular to v; in the diagram it is shown by w. If we draw the vector 
w at the point P it will obviously be directed towards the centre 
Oof the circle. 




Fig. 4. 



12 PARTICLE MECHANICS |I 

Thus the acceleration of a particle moving uniformly in a 
circle is towards the centre of the circle, i.e. at right angles to the 
velocity. 

Let us now determine the magnitude of the acceleration w. 
To do so, we must find the velocity of the point A moving in a 
circle of radius v. When P moves once round the circle about O, 
in a time T, say, the point A traverses the circle about C, a 
distance 2nv. The velocity of A is therefore 

w = 2ttvIT. 

Substituting the period T = Inr/v, where r is the radius of the 
path of the particle P, we obtain finally 

w = v 2 lr. 

Thus, if the velocity varies only in magnitude, the acceleration 
is parallel to the velocity; if the velocity varies only in direction, 
the acceleration and velocity vectors are mutually perpendicular. 

In general, when the velocity varies in both magnitude and 
direction, the acceleration has two components, one parallel to 
the velocity and one perpendicular to it. The first component, 
called the tangential component, is equal to the time derivative 
of the magnitude of the velocity: 

w t = dvjdt. 

The second component, w n , is called the normal component of the 
acceleration. It is proportional to the square of the velocity of the 
particle and inversely proportional to the radius of curvature of 
the path at the point considered. 

§7. Force 

If a particle is in free motion, i.e. does not interact with sur- 
rounding bodies, its momentum is conserved. If, on the other 
hand, the particle interacts with surrounding bodies, then its 
momentum varies with time. We can therefore regard the change 
in momentum of a particle as a measure of the action of sur- 
rounding bodies on it. The greater the change (per unit time), the 
stronger the action. It is therefore reasonable to take the time 



§7] FORCE 13 

derivative of the momentum vector of the particle in order to 
define this action. The time derivative is called the force on the 
particle. 

This definition describes one aspect of the interaction: it 
concerns the extent of the "reaction" of the particle to the action 
of surrounding bodies on it. Conversely, by studying the inter- 
action of the particle with surrounding bodies, we can relate the 
strength of this interaction to quantities describing the state of the 
particle and that of the surrounding bodies. 

The forces of interaction between particles depend (in classical 
mechanics) only on their position. In other words, the forces 
acting between particles depend only on the distances between 
them and not on their velocities. 

The manner in which the forces depend on the distances 
between the particles can in many cases be established by an 
examination of the physical phenomena underlying the interaction 
between particles. 

Let F denote the force acting on a given particle, expressed as 
a function of its coordinates and of quantities representing the 
properties and positions of the surrounding bodies. We can then 
write down an equation between two expressions for the force: 
F, and the change in the momentum p of the particle per unit time, 

dpi dt = F. 

This is called the equation of motion of the particle. 

Since p = my, the equation of motion of a particle may also 
be written 

m d\ldt = F. 

Thus the force acting on a particle is equal to the product of 
its acceleration and its mass. This is Newton's second law. 

It should be emphasised, however, that this law acquires a 
specific significance only when F is known as a function of the 
particle coordinates. In that case, i.e. if the form of the function 
F is known, the equation of motion enables us, in principle, to 
determine the velocity and coordinates of the particle as functions 
of time; that is, to find its path. In addition to the form of the func- 
tion F (i.e. the law of interaction between the particle and 



14 PARTICLE MECHANICS [i 

surrounding bodies), the initial conditions must be given, that is, 
the position and velocity of the particle at some instant taken as 
the initial instant. Since the equation of motion determines the 
increment of velocity of the particle in any time interval dt 
(d\ = F dtlm), and the velocity gives the change in spatial posi- 
tion of the particle (dr = v dt), it is clear that specifying the initial 
position and initial velocity of the particle is in fact sufficient to 
determine its further motion completely. This is the significance 
of the statement made in §2 that the mechanical state of a particle 
is defined by its coordinates and velocity. 

The equation of motion is a vector equation, and may therefore 
be written as three equations relating the components of accelera- 
tion and force: 



m 



dvjdt = F x , m dvyldt = F y , m dvjdt = F z 



Let us now consider a closed system of particles. As we 
know, the sum of the momenta of such particles is conserved: 

Pi + P2 + * ' ' = constant, 

where p t is the momentum of the ith particle. Differentiation of 
this equation with respect to time gives 

^1 + ^2+... = . 

dt dt 
Since 

dpjdt = F„ 
where F< is the force on the rth particle, we have 

F! + F 2 +- • =0. 

Thus the sum of all the forces in a closed system is zero. 

In particular, if the closed system contains only two bodies, 
the force exerted by one body on the other must be equal in 
magnitude and opposite in direction to the force which the latter 
body exerts on the former. This is called the law of action and 



§8] DIMENSIONS OF PHYSICAL QUANTITIES 15 



F, F 



Mi o ■ « 0M2 

Fig. 5. 

reaction or Newton's third law. Since, in this case, there is only 
one distinctive direction, namely that of the line joining the 
bodies (or particles), the forces Fj and F 2 must act along this line 
(see Fig. 5, where M x and M 2 denote the two particles). 

§ 8 . Dimensions of physical quantities 

All physical quantities are measured in certain units. To 
measure a quantity is to determine its ratio to another quantity 
of the same kind which is arbitrarily taken as the unit. 

In principle, any unit may be chosen for each physical quantity, 
but by using the relations between different quantities it is pos- 
sible to define a limited number of arbitrary units for certain 
quantities taken as fundamental, and to construct for the other 
quantities units which are related to the fundamental units. These 
are called derived units. 

Length, time and mass are taken as the fundamental quantities 
in physics. 

The unit of length in physics is the centimetre (cm), equal to 
one-hundredth of the metre, which is now defined as equal to 
1 650763-73 wavelengths of the light corresponding to a partic- 
ular (orange) line in the spectrum of the gas krypton. 

The metre was originally defined as one ten-millionth of a 
quadrant of the meridian through Paris, and the standard metre 
was constructed from measurements made in 1792. Since it was 
extremely difficult to reproduce the standard metre on the basis 
of its "natural" definition, the metre was later defined by agree- 
ment as the length of a particular standard — a platinum-iridium 
bar preserved at the International Bureau of Weights and 
Measures in Paris. This definition of the metre as a "distance 
between lines" has also now been abandoned, and the "light" 
metre described above is used. In consequence, the unit of length 
is again a natural and indestructible measure of length, and, 
moreover, allows a hundredfold increase in the accuracy of 
reproduction of the standard metre. 

The following units are used in measuring short distances: 
the micron (1^= 10~ 4 cm), the millimicron (lm/i= 10 _7 cm), 
the angstrom (1 A = 10 -8 cm) and the fermi (10 -13 cm). 



16 PARTICLE MECHANICS [i 

In astronomy, distances are measured in terms of the light- 
year, the distance traversed by light in one year, equal to 
9-46 x 10 17 cm. A distance of 3-25 light-years or 3-08 X 10 18 cm 
is called a par sec; it is the distance at which the diameter of the 
Earth's orbit subtends an angle of one second of arc. 

Time in physics is measured in seconds. The second (sec) is 
now defined as a certain fraction of a particular tropical year 
(1900). The tropical year is the time between successive passages 
of the Sun through the vernal equinox. The year (1900) is 
specified because the length of the tropical year is not constant 
but decreases by about 0-5 sec per century. 

The second was originally defined as a fraction (1/86 400) of 
the solar day, but the Earth's daily rotation is not uniform and the 
length of the day is not constant. The relative fluctuations of the 
length of the day are about 10~ 7 , which is too great for the day 
to be used as a basis for the definition of the unit of time, in 
terms of present-day technology. The relative fluctuations in the 
length of the tropical year are smaller, but the definition of the 
second on the basis of the Earth's revolution round the Sun 
cannot be regarded as entirely satisfactory, since it does not 
allow a "standard" unit of time to be reproduced with sufficient 
accuracy. This difficulty disappears only if the definition of the 
second is based not on the motion of the Earth but on the periodic 
motions occurring in atoms. The second then becomes a natural 
physical unit of time just as the "light" centimetre is a natural 
unit of length. 

Mass in physics is measured in grams, as already stated. One 
gram (g) is one-thousandth of the mass of a standard kilogram pre- 
served at the International Bureau of Weights and Measures in 
Paris. 

The mass of one kilogram was originally defined as the mass 
of one cubic decimetre of water at 4°C, i.e. the temperature at 
which water has its maximum density. It was, however, impos- 
sible to maintain this definition, as with the original definition of 
the metre, owing to the increasing accuracy of measurements; 
if the original definitions were retained it would be necessary to 
keep changing the fundamental standards. Modern results show 
that 1 cm 3 of distilled water at 4°C weighs not 1 g but 0-999 972 g. 

The definition of the kilogram as the mass of a standard, how- 
ever, suffers from the same defects as the definition of the metre 



§8] DIMENSIONS OF PHYSICAL QUANTITIES 17 

as a "distance between lines". The best procedure would be to 
define the gram not in terms of the mass of a standard kilogram 
but in terms of the mass of an atomic nucleus, such as the proton. 

Let us now see how derived units are constructed, taking a 
few examples. 

As the unit of velocity we could take any arbitrary velocity 
(for instance, the mean velocity of the Earth round the Sun, or 
the velocity of light), and refer all other velocities to this as the 
unit; but we can also use the definition of velocity as the ratio of 
distance to time and take as the unit of velocity the velocity at 
which a distance of one centimetre is traversed in one second. 
This velocity is denoted by 1 cm/sec. The symbol cm/sec is 
called the dimensions of velocity in terms of the fundamental 
units, the centimetre for length and the second for time. The 
dimensions of velocity are written 

[v] = cm/sec. 

The situation is similar for acceleration. The unit of accelera- 
tion could be taken to be any acceleration (for instance, that of 
a freely falling body), but we can use the definition of acceleration 
as the change of velocity per unit time, and take as the unit of 
acceleration the acceleration such that the velocity changes by 
1 cm/sec in one second. The notation for this unit is 1 cm/sec 2 , 
and the symbol cm/sec 2 denotes the dimensions of acceleration, 
written as 

[w] = cm/sec 2 . 

Let us now determine the dimensions of force and establish 
the unit of force. To do so, we use the definition of force as the 
product of mass and acceleration. Using square brackets to 
denote the dimensions of any physical quantity, we obtain for 
the dimensions of force the expression 

[F] = [m][w] = g.cm/sec 2 . 

As the unit of force we can take 1 g.cm/sec 2 , which is called one 
dyne. This is the force which gives a mass of 1 g an acceleration 
of 1 cm/sec 2 . 



18 PARTICLE MECHANICS |I 

Thus, by using the relations between various quantities, we 
can choose units for all physical quantities by starting from a 
small number of fundamental quantities whose units are chosen 
arbitrarily. The system of physical units with the centimetre, 
gram and second as the fundamental units of mass, length and 
time is called the physical or CGS system of units. 

It should not be thought, however, that the use of just three 
arbitrary fundamental units in this system has any deep physical 
significance. It arises only from the practical convenience of the 
system constructed from these units. In principle, a system of 
units could be constructed with any number of arbitrarily chosen 
units (see §22). 

Operations with dimensions are carried out as if the latter 
were ordinary algebraic quantities, i.e. they are subject to the 
same operations as numbers. The dimensions of both sides of 
any equation containing different physical quantities must ob- 
viously be the same. This fact should be remembered in checking 

formulae. 

It is often known from physical considerations that a particular 
physical quantity can depend only on certain other quantities. 
In many cases dimensional arguments alone suffice for the nature 
of the dependence to be determined. We shall later see examples 
of this. 

Besides the CGS system of units, other systems are frequently 
used, in which the fundamental units of mass and length are 
greater than the gram and the centimetre. The international 
system of units (SI) is based on the metre, kilogram and second 
as units of length, mass and time. The unit of force in this system 
is called the newton (N): 

1 N = 1 kg.m/sec 2 = 10 5 dyn. 

In engineering calculations, force is usually measured in units 
of kilogram-force (kgf). This is the force with which a mass of 
1 kg is attracted to the Earth at sea level in latitude 45°. Its value is 

1 kgf = 9-8 X 10 5 dyn = 9-8 N 

(more precisely, 980 665 dyn). 



§9] MOTION IN A UNIFORM FIELD 19 

§9. Motion in a uniform field 

If a particle is subject to a definite force at every point in 
space, these forces as a whole are called a force field. In general, 
the field forces may vary from one point to another in space and 
may also depend on time. 

Let us consider the simple case of the motion of a particle in a 
uniform and constant field, where the field forces have the same 
magnitude and direction everywhere and are independent of 
time, for example the Earth's gravitational field in regions small 
compared with its radius. 

From the equation of motion of a particle, 

m d\/dt = F, 

we have when F is constant 

v = (F//n)f+v , 

where v is the initial velocity of the particle. Thus in a uniform 
and constant field the velocity is a linear function of time. 

The expression obtained for v shows that the particle moves 
in the plane defined by the force vector F and the initial velocity 
vector v . Let us take this as the xy plane, and the y axis in the 
direction of the force F. The equation for the velocity v of the 
particle gives two equations for the velocity components v x 
and v y \ 

v y = (Flm)t + v y0 , v x = v x0 , 

where v x0 and v y0 are the initial values of the velocity components. 
Since the velocity components are the time derivatives of the 
corresponding coordinates of the particle, we can write the last 
two equations as 



Hence 



dyldt = (Flm)t + v y0 , dx/dt = v x0 . 
y = (Fl2m)t 2 + v y0 t + y , 

X = V x $t + Xq, 



20 PARTICLE MECHANICS [i 

where jc„ and y are the initial values of the coordinates of the 
particle. These expressions determine the path of the particle. 
They can be simplified if time is measured from the instant at 
which the velocity component v y is zero; then v y0 = 0. Taking the 




vo 
Fig. 6. 

origin at the point where the particle is at that instant, we have 
Xo = y = o. Finally, denoting the quantity v x0 , which is now the 
initial magnitude of the velocity, by v simply, we have 

y=(Fl2m)t 2 , x=v t. 

Elimination of t gives 

y = (F/2mu 2 )^, 

the equation of a parabola (Fig. 6). Thus a particle in a uniform 
field describes a parabola. 

§ 1 0. Work and potential energy 

Let us consider the motion of a particle in a force field F. If 
the particle moves an infinitesimal distance ds under the action 
of the force F, the quantity 

dA = F ds cos 0, 

where is the angle between the vectors F and ds, is called the 
work done by the force F over the distance ds. The product of 
the magnitudes of two vectors a and b and the cosine of the angle 
between them is called the scalar product of these vectors and 
denoted by a.b. The work may therefore be defined as the scalar 
product of the force vector and the particle displacement vector: 

dA = F.ds. 



§10] WORK AND POTENTIAL ENERGY 21 

This expression may also be written 

dA = F s ds, 

where F s is the component of the force F in the direction of 
motion of the particle. 

In order to determine the work done by field forces over a 
finite path of the particle, it is necessary to divide this path into 
infinitesimal intervals ds, find the work for each such interval, and 
add the results. The sum gives the work done by the field forces 
over the whole path. 

From the definition of work it follows that a force perpendicu- 
lar to the path does no work. In particular, in uniform motion of a 
particle in a circle the work done by forces is zero. 

A constant force field, i.e. one independent of time, has the 
following remarkable property: if a particle moves along a closed 
path in such a field, so as to return to its original position, then the 
work done by the field forces is zero. 




b 
Fig. 7. 



From this property there follows another result: the work done 
by the field forces in moving a particle from one position to an- 
other is independent of the path taken, and is determined only by 
the initial and final points. For let us consider two points 1 and 
2 joined by two curves a and b (Fig. 7), and suppose that the 
particle moves from point 1 to point 2 along curve a and then 
from point 2 back to point 1 along curve b. The total work done 
by the field forces during this process is zero. Denoting the work 
by^4, we can write 

^*la2 +^261 = 0. 

When the direction of motion is reversed, the sign of the work is 
obviously changed, and thus we have 



A\a2. — ^261 — A 



1&2? 



22 PARTICLE MECHANICS [l 

i.e. the work is independent of the form of the curve joining the 
initial and final points 1 and 2. 

Since the work done by the field forces is independent of the 
path taken and is determined only by the terminal points of the 
path, it is clearly a quantity of deep physical significance. It can 
be used to define an important property of the force field. To do 
so, we take any point O in space as the origin, and consider the 
work done by the field forces when the particle moves from O 
to any point P, denoting this work by — U. The quantity U, 
i.e. minus the work done in moving the particle from O to P, is 
called the potential energy of the particle at the point P. It is a 
function of the coordinates x, y, z of the point P: 

U=U(x,y,z). 

The work A 12 done by the field forces when the particle moves 
between any points 1 and 2 is 

A n = U t — U 2 , 

where U t and U 2 are the values of the potential energy at the two 
points. The work done is equal to the difference of the potential 
energies at the initial and final points of the path. 

Let us consider two points P and P' an infinitesimal distance 
apart. The work done by the field forces when the particle moves 
from P to P' is —dU. This work is also equal to F.ds, where ds is 
the vector from P to P' ; it has been shown in §2 that the vector 
ds is equal to the difference dv of the radius vectors of P' and P. 
Thus we obtain the equation 

Y.dv = -dU. 

This relation between the force and the potential energy is one of 
the fundamental relations of mechanics. 

Writing Y.dr = F.ds = F s ds, we can put this relation in the form 

F s = -dU/ds. 

This means that the component of the force in any direction is 
obtained by dividing the infinitesimal change dU in the potential 



§11] THE LAW OF CONSERVATION OF ENERGY 23 

energy over an infinitesimal interval in that direction by the length 
ds of the interval. The quantity dUlds is called the derivative of 
U in the direction s. 

To explain these relations, let us determine the potential energy 
in a constant uniform field. We take the direction of the field 
force F as the z axis. Then F.c/r = Fdz; equating this to the 
change in the potential energy, we have - dU = Fdz, whence 

U = —Fz + constant. 

We see that the potential energy is defined only to within an 
arbitrary constant. This is a general result related to the arbi- 
trariness of the choice of the original point O in the field from 
which the work done on the particle is measured. It is usual to 
choose the arbitrary constant in the expression for U so that the 
potential energy of the particle is zero when it is at an infinite 
distance from other bodies. 

From the relations between the force components and the 
potential energy we can deduce the direction of the force. If the 
potential energy increases in a given direction (dU/dt > 0), the 
component of the force in that direction is negative, i.e. the force 
is in the direction of decreasing potential energy. Force always 
acts in the direction in which potential energy decreases. 

Since the derivative vanishes at points where the function has a 
maximum or minimum, the force is zero at points of maximum or 
minimum potential energy. 

§11. The law of conservation of energy 

The fact that the work done by the forces of a constant field 
when a particle moves from one point to another is independent 
of the shape of the path along which the particle moves leads to 
an extremely important relationship, the law of conservation of 
energy. 

In order to derive this, we recall that the force F acting on the 
particle is 

F = md\/dt. 

Since the component of acceleration in the direction of the motion 
is dv/dt, the force component in this direction is 

F s = m dvldt. 



24 PARTICLE MECHANICS LI 

Let us now determine the work done by this force over an 
infinitesimal distance ds = v dt: 

dA = F s ds = mvdv, 

or 

dA = d^mv 2 ). 

Thus the work done by the force is equal to the increase in \nv?. 
This quantity is called the kinetic energy of the particle. 

The work is also equal to the decrease in potential energy: 
dA = —dU. We can therefore write 

-dU = d(imrf), 

i.e. 

d(U + %mv 2 ) = 0. 

Denoting the sum by E, we hence obtain 

E = Imu 2 +U = constant. 

Thus the sum of the kinetic energy of the particle, which 
depends only on its velocity, and the potential energy, which 
depends only on its coordinates, is constant during the motion 
of the particle. This sum is called the total energy or simply the 
energy of the particle, and the relationship derived above is 
called the law of conservation of energy. 

The force field in which the particle moves is generated by 
various other bodies. If the field is constant, these bodies must 
be at rest. Thus we have derived the law of conservation of energy 
in the simple case where one particle moves and all the other 
bodies with which it interacts are at rest. But the law of conserva- 
tion of energy can also be stated in the general case where more 
than one particle is moving. If these particles form a closed 
system, a law of conservation of energy is again valid which 
states that the sum of the kinetic energies of all the particles 



§11] THE LAW OF CONSERVATION OF ENERGY 25 

separately and their mutual potential energy does not vary with 
time, i.e. 

E = im 1 v 1 2 + ^m 2 v 2 2 + h U(r u r 2 ,. . . ), 

where m t is the mass of the ith particle, \ t its velocity and U 
the potential energy of interaction of the particles, which depends 
on their radius vectors r t . 

The function U is related to the forces acting on each particle 
in the same way as for a single particle in an external field. In 
determining the force F, acting on the rth particle we must con- 
sider the change in the potential energy U in an infinitesimal 
displacement dr t of this particle, the positions of all the other 
particles remaining unchanged. The work F,.dr, done on the 
particle in such a displacement is equal to the corresponding 
decrease in the potential energy. 

The law of conservation of energy is valid for any closed 
system and, like the law of conservation of momentum, is one of 
the most important laws of mechanics. 

The kinetic energy is an essentially positive quantity. The 
potential energy of interaction of particles may be either positive 
or negative. If the potential energy of two particles is so defined 
that it is zero when the particles are at a great distance apart, 
its sign depends on whether the interaction between the particles 
is attractive or repulsive. Since the forces acting on particles 
are always in the direction of decreasing potential energy, the 
approach of attracting particles leads to a decrease in potential 
energy, which is therefore negative. The potential energy of 
repelling particles, on the other hand, is positive. 

Energy, and also work, have the dimensions 

[E] = [m][i?] = g.cm 2 /sec 2 . 

The unit of energy in the CGS system is therefore 1 g.cm 2 /sec 2 , 
which is called the erg. It is the work done by a force of 1 dyn 
acting through 1 cm. 

In the SI system a larger unit of energy, the joule (J), is used, 
equal to the work done by a force of 1 N acting through 1 m: 

1 J= 1 N.m= 10 7 erg. 



26 PARTICLE MECHANICS [i 

If the unit of force is the kilogram-force, the corresponding 
unit of energy is the kilogram-metre (kgf.m), equal to the work 
done by a force of 1 kgf acting through 1 m. It is related to the 
joule by 1 kgf.m = 9-8 J. 

Energy sources are described by the work done per unit time. 
This is called the power. The unit of power is the watt (W): 

1 W = 1 J/sec. 

The work done in one hour by an energy source of power 1 W is 
called a watt-hour (Wh). It is easy to see that 

lWh^^xlO 3 .!. 

§12. Internal energy 

As has been explained in §5, for the motion of a composite 
system we can define the velocity of the system as a whole, 
namely the velocity of the centre of mass of the system. This 
means that the motion of the system may be regarded as con- 
sisting of two parts: the motion of the system as a whole and the 
"internal" motion of the particles forming the system relative 
to the centre of mass. Accordingly the energy E of the system may 
be written as the sum of the kinetic energy of the system as a 
whole, which is WV 2 (where M is the mass of the system and V 
the velocity of its centre of mass), and the internal energy E int of 
the system, which comprises the kinetic energy of the internal mo- 
tion of the particles and the potential energy of their interaction: 

E = WV 2 + E m . 

Although this formula is fairly obvious, we shall also give a 
direct derivation of it. The velocity of the /th particle, say, 
relative to a fixed frame of reference may be written as Vj + V, 
where V is the velocity of the centre of mass of the system and 
v, is the velocity of the particle relative to the centre of mass. 
The kinetic energy of the particle is 

Imfa + V) 2 = im, V 2 + im t v t 2 + m t V . v,. 

On summation over all particles the first term from each such 
expression gives \MV 2 , where M = m 1 + m 2 +- • • . The sum of 



§13] BOUNDARIES OF THE MOTION 27 

the second terms gives the total kinetic energy of the internal 
motion in the system. The sum of the third terms is zero, since 

m t \ . Vx + maV. v 2 + - • • = V.(ra 1 v 1 +ra 2 v 2 + - • •); 

the expression in parentheses is the total momentum of the 
particles relative to the centre of mass of the system, which by 
definition is zero. Finally, adding the kinetic energy to the poten- 
tial energy of interaction of the particles, we obtain the required 
formula. 

Using the law of conservation of energy, we can discuss the 
stability of a composite body. The problem here is to ascertain 
the conditions in which the composite body may spontaneously 
disintegrate into its component parts. Let us consider, for 
example, the break-up of a composite body into two parts; let 
the masses of the parts be m l and m*, and let the velocities of 
the parts in the centre-of-mass frame of the original composite 
body be v 2 and v 2 . Then the law of conservation of energy in this 
frame is 



£int = \rri\V? + E lint + $m2V 2 2 + E. 



2int» 



into 



where £ int is the internal energy of the original body and E Uu „ 
E 2iat the internal energies of the two parts. Since the kinetic 
energy is always positive, it follows from the above relation that 

^int > ^lint+^int- 

This is the condition for the body to be able to disintegrate into 
two parts. If, on the other hand, the internal energy of the body is 
less than the sum of the internal energies of its component parts, 
the body will be stable with respect to the disintegration. 

§13. Boundaries of the motion 

If the motion of a particle is constrained so that it can move 
only along a certain curve, the motion is said to have one degree 
of freedom or to be one-dimensional. One coordinate is then 
sufficient to specify the position of the particle; it may be taken, 
for example, as the distance along the curve from a point taken as 
origin. Let this coordinate be denoted by x. The potential energy 



28 



PARTICLE MECHANICS 



of a particle in one-dimensional motion is a function only of this 
one coordinate: U = U(x). 
According to the law of conservation of energy we have 

E = \m& + U(x) = constant, 

and since the kinetic energy cannot take negative values the 
inequality 

U ^E 

must hold. This implies that the particle during its motion can 
occupy only points where the potential energy does not exceed 
the total energy. If these energies are equal, we have the equation 

U(x) = E, 
which determines the limiting positions of the particle. 



iU(x) 




Fig. 8. 



Some typical examples are the following. Let us first take a 
potential energy which, as a function of the coordinate x, has the 
form shown in Fig. 8. In order to find the boundaries of the 
motion of a particle in such a force field, as functions of the total 
energy E of the particle, we draw a straight line U = E parallel 
to the x axis. This line intersects the curve of potential energy 
U = u( x ) a t two points, whose abscissae are denoted by x t and 
jc 2 . If the motion is to be possible it is necessary that the potential 
energy should not exceed the total energy. This means that the 
motion of a particle with energy E can occur only between the 
points x x and x 2 , and a particle of energy E cannot enter the 
regions right of x 2 and left of x x . 



§13] BOUNDARIES OF THE MOTION 29 

A motion in which the particle remains in a finite region of 
space is called a finite motion; one in which the particle can go to 
any distance is called an infinite motion. 

The region of finite motions depends, of course, on the energy; 
in the example considered here, it decreases with decreasing 
energy and shrinks to a single point x when E = U min . 

At the points Xj and x 2 the potential energy is equal to the total 
energy, and therefore at these points the kinetic energy and hence 
the particle velocity are zero. At the point jc the potential energy 
is a minimum, and the kinetic energy and velocity have their 
maximum values. Since the force F is related to the potential 
energy F = —dUldx, it is negative between jc and x 2 , and positive 
between * and x x . This means that between jc and x 2 the force is 
in the direction of decreasing x, i.e. to the left, and between x and 
x t it is to the right. Consequently, if the particle begins to move 
from the point x u where its velocity is zero, the force to the right 
will gradually accelerate it to a maximum velocity at the point x Q . 
As the particle continues to move from jc to x 2 under the force 
which is now to the left, it will slow down until it comes to rest 
at x 2 . It will then begin to move back from x 2 to x . This type of 
motion will continue indefinitely. Thus the particle executes a 
periodic motion with a period equal to twice the time for the 
particle to go from x t to x 2 . 

At the point x the potential energy is a minimum and the 
derivative of U with respect to x is zero; at this point the force is 
therefore zero, and the point jc is consequently a position of 
equilibrium of the particle. This position is evidently one of 
stable equilibrium, since in this case a departure of the particle 
from the equilibrium position causes a force which tends to 
return the particle to the equilibrium position. This property 
exists only for minima and not for maxima of the potential 
energy, although at the latter the force is likewise zero. If a par- 
ticle is moved in either direction from a point of maximum poten- 
tial energy, the resulting force in either case acts away from this 
point, and points where the potential energy reaches a maximum 
are therefore positions of unstable equilibrium. 

Let us now consider the motion of a particle in a more complex 
field whose potential-energy curve has the form shown in Fig. 9. 
This curve has both a minimum and a maximum. If the particle 
has energy E, it can move in such a field in two regions: region I 



30 



PARTICLE MECHANICS 



between the points Xi and x 2 , and region III to the right of the 
point x A (at these points the potential energy is equal to the total 
energy). The motion in the former region is of the same type as in 
the previous example, and is oscillatory. The motion in region III, 
however, is infinite, since the particle may move to any distance 
to the right of the point jc 3 . If the particle begins its motion at the 
point * 3 , where its velocity is zero, it will continually be accele- 
rated by the force to the right; at infinity, the potential energy is 
zero and the particle velocity reaches the value v„ = V(2mE). 



U(x) 




Fig. 9. 

If, on the other hand, the particle moves from infinity to the point 
x 3 , its velocity will gradually decrease and vanish at x s , where the 
particle will turn round and go back to infinity. It cannot pene- 
trate into region I, since this is prevented by the forbidden region 
II lying between x 2 and x 3 . This region also prevents a particle 
that is executing oscillations between *i and x 2 from entering 
region III, where motion with energy E is also possible. The for- 
bidden region is called a potential barrier, and region I is called a 
potential well As the particle energy increases in this case, the 
width of the barrier diminishes and for E 2= U max it does not exist. 
The region of oscillatory motion likewise disappears, and the 
motion of the particle becomes infinite. 

Thus we see that the motion of a particle in a given force field 
may be either finite or infinite depending on the energy of the 
particle. 

This may be illustrated also by the example of motion in a field 
whose potential-energy curve has the form shown in Fig. 10. In 
this case positive energies correspond to infinite motion, and 
negative energies (U min < E < 0) to finite motion. 



§14] 



ELASTIC COLLISIONS 



31 



U(x) i 




Fig. 10. 

Whenever the potential energy is zero at infinity, motion with 
negative energy will necessarily be finite, since at infinity the 
zero potential energy exceeds the total energy, and the particle 
therefore cannot go to infinity. 

§14. Elastic collisions 

The laws of conservation of energy and momentum can be 
used to establish relations between various quantities in collisions. 

In physics, collisions are processes of interaction between 
bodies in the broad sense of the word, and do not necessarily 
involve literal contact between the bodies. The colliding bodies 
are free when at an infinite distance apart. As they pass they 
interact, and in consequence of this various processes may 
occur: the bodies may combine, may form new bodies or may 
undergo an elastic collision, in which the bodies move away after 
their approach, without any change in their internal state. 
Collisions in which a change occurs in the internal state of the 
bodies are said to be inelastic. 

Collisions between ordinary bodies under ordinary conditions 
are almost always inelastic to some extent, if only because they are 
accompanied by some heating of the bodies, that is, by the con- 
version of part of their kinetic energy into heat. Nevertheless, the 
concept of elastic collisions is of great importance in physics, 
since such collisions are often involved in physical experiments 
dealing with atomic phenomena. Ordinary collisions also may 
frequently be regarded as elastic to a sufficient approximation. 



32 PARTICLE MECHANICS [l 

Let us consider an elastic collision between two particles of 
masses m t and ra^ let their velocities before and after the collision 
be respectively V!, v 2 ; v/, v 2 '. We shall suppose that the particle 
AMg is at rest before the collision, i.e. v 2 = 0. 

Since, in an elastic collision, the internal energies of the 
particles are unchanged, they can be ignored in applying the 
law of conservation of energy, i.e. they can be taken as zero. 
Since the particles are assumed not to interact before and after 
the collision, i.e. to be free, the law of conservation of energy 
amounts to the conservation of kinetic energy: 

mjfi 2 = m 1 v 1 ' 2 + m 2 v 2 ' 2 , 

where the common factor £ has been omitted. 

The law of conservation of momentum is expressed by the 
vector equation 

m^! = m x \x +171^2 . 

A very simple case is that where the mass of the particle 
originally at rest is much greater than that of the incident particle, 
i.e. ra 2 > m x . The formula 

v 2 ' = (m 1 /m 2 )(v 1 -v 1 ') 

shows that the velocity v 2 ' will then be very small. A similar 
conclusion may be drawn regarding the energy of this particle 
originally at rest, since the product m^' 2 will be inversely 
proportional to the mass m^. Hence we deduce that the energy 
of the first (incident) particle is unchanged by the collision, and 
its velocity is therefore unchanged in magnitude. Thus a collision 
between a light and a heavy particle can change only the direction 
of the velocity of the light particle, the magnitude of its velocity 
remaining constant. 

If the masses of the colliding particles are equal, the conserva- 
tion laws become 

Vi = Vx'+V 2 ', 

v 1 2 = v 1 ' 2 + v 2 ' 2 . 



§ 14 ] ELASTIC COLLISIONS 33 

The first of these relations signifies that the vectors v l5 v/ and 
v 2 ' form a triangle; the second shows that the triangle is right- 
angled with hypotenuse v x . Thus two particles of equal mass 
diverge at right angles after the collision (Fig. 1 1). 




Let us next consider a head-on collision of two particles. After 
such a collision the two particles will move along the direction 
of the velocity of the incident particle. In this case we can replace 
the velocity vectors in the law of conservation of momentum by 
their magnitudes: 

WaV = rrixivx — Vx). 

Using also the law of conservation of energy, according to which 

m 2 v 2 ' 2 = m 1 {v 1 2 -v 1 ' 2 ), 

we can express v x ' and v 2 ' in terms of v t . Dividing the second 
expression by the first gives 

V = Vi+Vi 

and therefore 

, _ mx-rrh 2m x 

r i ~ M _L „ v u v 2 ' = 



mj + mg " m 1 +-m i 1 ' 

The first (incident) particle will continue to move in the same 
direction or will move back in the opposite direction, according 
as its mass m x is greater or less than the mass m, of the particle 
originally at rest. If the masses m 1 and m 2 are equal, then V = 0, 
v 2 ' = v u so that the particles as it were exchange velocities. If 
nh, > m u then v x ' = — Vl and v 2 = 0. 

In the general case it is convenient to consider the collision in 
the centre-of-mass frame of the colliding particles. Then the total 



34 PARTICLE MECHANICS U 

momentum of the particles is zero both before and after the 
collision. Hence, if the momenta of the first particle before and 
after the collision are p and p', those of the second particle will 
be — p and — p' respectively. 

Next, equating the sums of the kinetic energies of the particles 
before and after the collision, we see that p? = p' 2 , i.e. the 
momenta of the particles are unchanged in magnitude. Thus the 
only effect of the collision is to rotate the momenta of the par- 
ticles, changing their direction but not their magnitude. The 
velocities of the two particles are changed in the same manner, 
being rotated without change of magnitude and remaining 
opposite in direction, as shown in Fig. 12; the suffix zero to the 
velocities is used to indicate that they are measured in the 
centre-of-mass frame. 




Fig. 12. 

The angle through which the velocities are turned is deter- 
mined not only by the laws of conservation of momentum and 
energy but also by the nature of the interaction between the 
particles and by their relative position in a collision. 

In order to ascertain how the velocities are changed in the 
original or laboratory frame of reference (in which one particle 
is at rest before the collision, i.e. v 2 = 0), we use the following 
graphical procedure. We construct a vector 01 equal to the 
velocity v 10 of the first particle in the centre-of-mass frame 
(Fig. 13). This velocity is related to the velocity \i of the same 
particle in the laboratory frame (which is also the relative 
velocity of the two particles) by v 10 = Vj — V, where 

_ m x y x + my\ 2 __ mxVx 
rrix + rrh m^rrh 

is the velocity of the centre of mass. Subtraction gives 

_ rrhj\i 
Vio — i 



§14] 



ELASTIC COLLISIONS 



35 




Fig. 13. 

The velocity v 10 ' of the first particle after the collision is obtained 
by turning the velocity v 10 through some angle 9, i.e. it may be 
represented by any radius OY of the circle in Fig. 13. To change 
to the laboratory frame of reference, we must add to all velocities 
the velocity V of the centre of mass. In Fig. 13 this is represented 
by AO. The vector A\ then gives the velocity \ x of the incident 
particle before the collision, and A 1 ' is the required velocity of 
that particle after the collision. The velocity of the second particle 
may be found similarly. 

In Fig. 13 it is assumed that m x < nti, so that the point A lies 
within the circle. The vector AY , i.e. the velocity v/, may have 
any direction. If m l > rrh, however, A lies outside the circle 
(Fig. 14). In this case the angle <f> between the velocities of the 
particle before and after the collision cannot exceed some maxi- 
mum value corresponding to A 1 ' being a tangent to the circle. The 




Fig. 14. 



36 PARTICLE MECHANICS [i 

side AY of the triangle AVO is then perpendicular to 01', and 

sin</> max = Ol'/AO = m^lm^ 

We may also note that the velocity of the particle after the 
collision cannot be less than a certain minimum value, which is 
reached when the point 1' in Fig. 13 (or Fig. 14) is diametrically 
opposite to 1. This corresponds to a head-on collision of the 
particles, and the minimum value of the velocity is 

, = Imx-mgl 

Vi mm mi + m2 v l' 

§15. Angular momentum 

Besides energy and momentum, another vector quantity called 
angular momentum is conserved for any closed system. This 
quantity is the sum of the angular momenta of the individual 
particles, defined as follows. 

Let a particle have momentum p and let its position relative to 
some arbitrary origin O be given by the radius vector r. Then the 
angular momentum L of the particle is defined as a vector whose 
magnitude is 

L= rp sin 6 

(where is the angle between p and r) and whose direction is 
perpendicular to the plane through the directions of p and r. The 
latter condition does not completely define the direction of L, 
since it may still be either "up" or "down". It is customary to 
define the direction of L as follows: if a right-handed screw is 
imagined to turn from the direction of r towards p, it will advance 
in the direction of L (Fig. 15). 




Fig. 15. 



§15] 



ANGULAR MOMENTUM 



37 



The quantity L may also be regarded in a more intuitive way if 
we note that the product r sin is the length h p of the perpendic- 
ular from O to the line of the particle momentum (Fig. 16); this 
distance is often called the moment arm of the momentum rela- 
tive to O. The angular momentum of the particle is equal to the 
product of this arm and the magnitude of the momentum: 

L = ph p . 

This vector L is simply the vector product defined in vector 
algebra; the vector L constructed in the manner described from 
the vectors r and p is called the vector product of r and p and 
written 



or, since p = my, 



L = rxp 



L = mr X v. 



This formula determines the angular momentum of a single 
particle. The angular momentum of a system of particles is 
defined as the sum of the individual angular momenta: 

L = 1-j X pj + r 2 X p 2 H . 

This sum is constant in time for any closed system -the law of 
conservation of angular momentum. 




Fig. 16. 



It should be noted that the definition of the angular momentum 
involves an arbitrarily chosen origin O from which the radius 
vectors of the particles are measured. Although the magnitude 



38 PARTICLE MECHANICS [i 

and direction of the vector L depend on the choice of O, it is 
easily seen that this dependence does not affect the law of con- 
servation of angular momentum. For if we move the point O 
through some distance a of given magnitude and direction, the 
radius vectors of the particles will all be changed by that amount, 
and the angular momentum is changed by 

axp! + axp 2 +- • • = ax(p! + p 2 +- • ) = aXP, 

where P is the total momentum of the system. For a closed 
system P is constant, and we therefore see that changing the ori- 
gin does not affect the constancy of the total angular momentum 
of a closed system. 

The angular momentum of a system of particles is usually 
defined with respect to the centre of mass of the system as origin. 
This will be assumed below. 

Let us determine the time derivative of the angular momentum 
of a particle. The rule for differentiation of a product gives 

dL d, , dr , dp 
-r = -r-(r X p) = -j- X p + r X -£ 
dt dr v dt v dt 

Since drldt is the velocity v of the particle, and p = rav, the first 
term is rav X v = 0, because the vector product of any vector 
with itself is zero. In the second term the derivative dp/dt is, as 
we know, the force F acting on the particle. Thus 

dL/dt = rXF. 

The vector product rXF is called the torque (relative to a 
given point O) and will be denoted by K: 

K = r X F. 

Similarly to the previous discussion of the angular momentum, we 
can say that the magnitude of the torque is equal to the product of 
the magnitude F of the force and its moment arm h F , i.e. the 
length of the perpendicular from O to the line of action of the 
force: 

K = Fh F . 



§15] ANGULAR MOMENTUM 39 

Thus the rate of change of the angular momentum of a particle 
is equal to the torque acting on it: 

dL/dt = K. 

The total angular momentum of a closed system is conserved; 
the time derivative of the sum of the angular momenta of the 
particles in the system is therefore zero: 

£u + i, + ...,-f + £ + ...-a 

Hence it follows that 

K 1 + K 2 + --. = 0. 

We see that in a closed system the sum of the torques is zero, 
as well as the sum of the forces on all the particles (§7). The 
latter statement is equivalent to the law of conservation of 
momentum, and the former to the law of conservation of angular 
momentum. 

There is a profound relation between these properties of a 
closed system and the fundamental properties of space itself. 

Space is homogeneous. This means that the properties of a 
closed system do not depend on its position in space. Let us 
suppose that a system of particles undergoes an infinitesimal 
displacement in space, whereby all the particles are moved the 
same distance in the same direction, and let the vector of this 
displacement be dR. The work done on the ith particle is F, . dR. 
The sum of the work done must be equal to the change in the 
potential energy of the system; but since the properties of the 
system do not depend on its position in space, this change must 
be zero. Thus we must have 

F x . dR + F 2 . dR+ • • • = (F! + F 2 + • • •)• dR = 0. 

Since this equation must hold for any direction of the vector dR, 

it follows that the sum of the forces Fj + F 2 H must be zero.' 

We therefore see that the source of the law of conservation 
of momentum is related to the property of homogeneity of 
space. 



40 PARTICLE MECHANICS [l 

A similar relation exists between the law of conservation of 
angular momentum and another fundamental property of space, 
its isotropy, i.e. the equivalence of all directions in space. As 
a result of this isotropy the properties of a closed system are 
unchanged when the system undergoes any rotation as a whole, 
and the work done in such a rotation is therefore zero. It can be 
shown that this condition leads to the vanishing of the sum of 
the torques in a closed system; we shall return to this topic 
in §28. 

§ 1 6. Motion in a central field 

The law of conservation of angular momentum is valid for a 
closed system, and not in general for the individual particles 
forming the system; but it may in fact be valid for a single par- 
ticle moving in a force field. For this to be so the field must be a 
central field. 

The term central field denotes a force field in which the poten- 
tial energy of a particle is a function only of its distance r from a 
certain point, the centre of the field: U=U(r). The force acting 
on a particle in such a field also depends only on the distance r 
and is along the radius from the centre to any point in space. 

Although a particle moving in such a field is not a closed system, 
the law of conservation of angular momentum is nevertheless 
valid for it if the angular momentum is defined relative to the 
centre of the field. For, since the line of action of the force 
acting on the particle passes through the centre of the field, 
the arm of the force about that point is zero, and the torque 
is therefore zero. From the equation dL/dt = K we then have 
L = constant. 

Since the angular momentum L = mr X v is perpendicular to 
the direction of the radius vector r, the constant direction of L 
shows that, as the particle moves, its radius vector must remain in 
one plane, perpendicular to the direction of L. Thus particles in 
a central field move in plane orbits, the plane of each orbit passing 
through the centre of the field. 

The law of conservation of angular momentum in such a "plane" 
motion may be put in an intuitive form. To do so, we write L in 
the form 

L = mr X v = mr X (ds/dt) = mr X ds/dt, 



§16] 



MOTION IN A CENTRAL FIELD 



41 



where ds is the vector of the displacement of a particle in a time 
dt. The magnitude of the vector product of two vectors has the 
geometrical significance of the area of the parallelogram which 
they form. The area of the parallelogram formed by the vectors 
ds and r is twice the area of the infinitely narrow sector OAA' 
(Fig. 17) swept out by the radius vector of the moving particle in 
time dt. Denoting this area by dS, we can write the magnitude of 
the angular momentum as 

L = 2m dS/dt. 

The quantity dS/dt is called the sectorial velocity. 

Thus the law of conservation of angular momentum can be 
formulated in terms of the constancy of the sectorial velocity: 
the radius vector of the moving particle describes equal areas in 
equal times. In this form it is called Kepler's second law. 




Fig. 17. 

The problem of motion in a central field is particularly impor- 
tant because the problem of the relative motion of two interacting 
particles (the two-body problem) can be reduced to it. 

Let us consider this motion in the centre-of-mass frame of the 
two particles. In this frame of reference the total momentum of 
particles is zero: 

m^i + m 2 \ 2 = 0, 

where v a and v 2 are the velocities of the particles. Let the relative 
velocity of the particles be 



V = V!-V 2 . 



42 PARTICLE MECHANICS [i 

From these two equations we easily find the formulae 

Vi = ; v, v 2 = — v, 

which express the velocity of each particle in terms of their 
relative velocity. 

We substitute these formulae in the expression for the total 
energy of the particles, 

E = ±m x v t 2 + \m 2 v 2 2 +U(r), 

where U{r) is the mutual potential energy of the particles as a 
function of the distance r between them (i.e. of the magnituue of 
the vector r = r x — r 2 ). A simple reduction then gives 

E = %rm? + U(r), 

where 

m = ntxmiKmx + m^) 

and is called the reduced mass of the particles. 

We see that the energy of the relative motion of the two 
particles is the same as if a single particle of mass m were moving 
with velocity v = dr/dt in a central external field with potential 
energy U(r). Thus the problem of the motion of two particles is 
equivalent to that of a single "reduced" particle in an external 

field. 

If the solution of the latter problem is known (i.e. if the path 
r = r(0 of the "reduced" particle has been found), we can 
immediately find the actual paths of the two particles m x and 
W2 by means of the formulae 

ma m x 

r > r 2 = — ^ ■ „ r ' 



rtix + rrh ' rrii + mz 



which express the radius vectors of the particles r x and r 2 with 
respect to their centre of mass in terms of their distance apart 
r = r x - r 2 ; these formulae follow from the relation m^ + m 2 r 2 = 



§16] MOTION IN A CENTRAL FIELD 43 

and correspond to the analogous formulae given above for the 
velocities v x = drjdt and v 2 = dr 2 /dt. Hence we see that the two 
particles will move relative to the centre of mass of the system 
along geometrically similar paths which differ only in having 
sizes inversely proportional to the masses of the particles: 

rjr 2 = nk/m!. 

During the motion the particles are always on a line passing 
through the centre of mass. 



CHAPTER II 

FIELDS 



§17. Electrical interaction 

In Chapter I we have given a definition of force and the 
relation between force and potential energy. We shall now go on 
to a specific analysis of some of the interactions underlying 
various physical phenomena. 

One of the most important kinds of interaction in Nature is 
electrical interaction. In particular, the forces acting in atoms 
and molecules are essentially of electrical origin, and this inter- 
action is therefore what mainly determines the internal structure 
of various bodies. 

The forces of electrical interaction depend on the existence of 
a particular physical characteristic of particles, their electric 
charge. Bodies having no electric charge have no electrical 
interaction. 

If bodies may be regarded as particles, the force of electrical 
interaction between them is proportional to the product of the 
charges on the bodies and inversely proportional to the square of 
the distance between them. This is called Coulomb's law. 
Denoting the electrical interaction force by F, the charges on 
the bodies by e x and e 2 , and the distance between them by r, we 
can write Coulomb's law in the form 

F = constant X e^e^r 2 . 

The force F acts along the line joining the charges, and experi- 
ment shows that it is sometimes an attraction, sometimes a 
repulsion. Charges are therefore said to differ in sign. Bodies 
having charges of the same sign repel each other, while bodies 
having charges of opposite signs attract each other. A positive 
sign of the force in Coulomb's law denotes repulsion, and a 
negative sign attraction. It does not matter which charges are 
in fact regarded as positive and which as negative, and the 

44 



§17] ELECTRICAL INTERACTION 45 

choice usual in physics is a historical convention. Only a differ- 
ence in the sign of charges has intrinsic significance. If all 
negative charges were called positive and vice versa, there 
would be no resulting change in the laws of physics. 

Since charges are now introduced for the first time and no 
units of charge have yet been defined, we can take the proportion- 
ality coefficient in Coulomb's law equal to unity: F = e t e 2 /r 2 . 
This establishes a unit of charge, namely the charge whose force 
of interaction with another similar charge at a distance of one 
centimetre is one dyne. This is called the electrostatic unit of 
charge. The system of units based on this choice of the constant 
coefficient in Coulomb's law is called the electrostatic or CGSE 
system. In this system the dimensions of charge are 

W=([F][/-] 2 ) 1 ' 2 

= (^Wj /2 =g 1 ' 2 cm 3 ' 2 sec- 1 . 

In the SI system of units a larger unit of charge is used, called 
the coulomb: 

1 coulomb = 1 C = 3 X 10 9 CGSE units of charge. 

By means of the expression for the force of electrical inter- 
action we can find the mutual potential energy of two electric 
charges e x and e 2 . If the distance between these charges increases 
by dr, the work done is dA = e x e 2 dr\r % . This is equal to the 
decrease in the potential energy U. Thus 

—dU = e 1 e 2 drlr 2 

= —e 1 e 2 d(llr) t 
whence 

U = e x e 2 lr. 

Strictly speaking, a constant term may also be included here; we 
have taken it as zero, in order that the potential energy should 
be zero when the charges are at an infinite distance apart. The 
potential energy of the interaction of two charges is therefore 
inversely proportional to the distance between them. 



46 FIELDS [II 

§18. Electric field 

Since Coulomb's law involves the product of the charges, the 
force exerted on a charge e by another charge e x can be put in 
the form 

F = eE, 

where E is a vector independent of the charge e and determined 
only by the charge e t and the distance r between the charges e 
and e x . This vector is called the electric field due to the charge 
e x . Its magnitude is 

E = ejr 2 

and it is directed along the line joining the positions of the charges 
e x and e. The force on e due to e x is thus the product of e and the 
electric field at e due to e x . 

Thus we have another way of describing electrical interaction. 
Instead of saying that particle 1 attracts or repels particle 2, we 
say that the first particle, whose electric charge is e lf creates a 
particular force field in the surrounding space, namely an electric 
field; particle 2 does not interact directly with particle 1, but is 
subject to the field created by the latter. 

These two ways of describing the interaction are presented 
here as being only formally different. In reality, however, this is 
not so; the concept of the electric field is by no means formal. An 
analysis of electric (and magnetic) fields which vary with time 
shows that they can exist in the absence of electric charges and 
are physically real in the same way as the particles that exist in 
Nature; however, such problems are outside the scope of the 
basic ideas concerning interactions of particles that are discussed 
here in connection with the laws of particle motion. 

The electric field created by not one but several electric charges 
is determined by the following fundamental property of electrical 
interactions: the electrical interaction between two charges is 
independent of the presence of a third charge. From this we can 
conclude that, if there are several charged particles, the electric 
field which they create is equal to the vector sum of the electric 
fields produced by each particle separately. In other words, the 
electric fields created by different charges are simply superposed 
without affecting one another. This remarkable property of the 
electric field is called the property of superposition. 



§18] ELECTRIC FIELD 47 

It should not be thought that the property of superposition of 
electric fields is a direct consequence of the existence of electrical 
interaction. In reality, this fundamental property of the electric 
field is a law of Nature. It applies to other fields besides electric 
fields and plays a very important part in physics. 

Let us apply the property of superposition to determine the 
electric field of a composite body at large distances from it. If 
the charges on the particles which compose the body are e x , e 2 , . . ., 
then the fields which they create at a distance r are 

Ei = e x lr 2 , E 2 = e 2 /r 2 , 

At large distances from the body we may regard the distances 
from the various particles as equal and the direction, from the 
particles to the point considered, as constant. Thus by using the 
property of superposition to find the total field E due to the body, 
we can simply take the algebraic sum of the fields E U E 2 ,...: 

E= (e 1 + e 2 +---)lr i . 

We see that the field of a composite body is the same as the field 
of a single particle with charge 

e = e 1 + e 2 +- - -. 

In other words, the charge on the composite body is equal to the 
sum of the charges on the particles which compose the body and 
does not depend on their relative position and motion. This is 
called the law of conservation of charge. 

In general the electric field is complicated, varying from point to 
point in both magnitude and direction. To represent it graphically 





Fig. 18. 



48 



FIELDS 



[II 



we can use electric lines of force; these are lines which at every 
point in space have the direction of the electric field acting at that 

point. 

If the field is created by a single charge, the lines of force are 
straight lines radiating from the position of the charge, or con- 
verging to its position, according as the charge is positive or 
negative (Fig. 18). 

From the definition of the lines of force it is clear that only 
one line of force passes through each point in space (not occupied 
by an electric charge), in the direction of the electric field acting 
at that point. Thus the lines of force do not intersect at points in 
space where there are no charges. 

Electric lines of force in a constant field cannot be closed. For 
when a charge moves along a line of force, the field forces do a 
positive amount of work, since the force is always along the path. 
If there existed closed lines of force, therefore, the work done by 
the field forces when a charge moved along such a line back to 
its starting point would not be zero, in conflict with the law of 
conservation of energy. 

Thus the lines of force must necessarily begin and end, or else 
go to infinity. The points where they begin and end are the charges 
which create the field. A line of force cannot go to infinity at both 
ends, since if it did, the field forces would do work when a charge 




Fig. 19. 



§19] ELECTROSTATIC POTENTIAL 49 

is transported along such a line from infinity and back to infinity, 
in contradiction with the fact that the potential energy is zero at 
both ends of the path. 

One end of a line of force must therefore necessarily be at a 
charge; the other may go either to infinity or to a charge of the 
opposite sign. As an illustration, Fig. 19 shows the field of two 
charges with opposite signs, +e x and -e 2 . The diagram is for the 
case where e x is greater than e 2 . Then some of the lines of force 
leaving +e t end at the charge ~e 2 , while the others go to infinity. 

§ 1 9. Electrostatic potential 

Like the force, the potential energy U of a charge e in an 
electric field is proportional to the magnitude of the charge, i.e. 

U=e<f). 

The quantity <f> which appears here, and which is the potential 
energy of a unit charge, is called the potential of the electric 
field 

On comparing this definition with that of the field (F = eE, 
where F is the force acting on the charge e) and using the general 
relation between force and potential energy, F s = -dU/ds 
(see §10), we find that a similar relation holds between the field 
and the potential: 

E s = —d<j>lds. 

The potential energy of two charges e x and e 2 at a distance r is, 
as we know, 

U = e x e 2 lr. 

The potential of the field due to a charge e x at a distance r is 
therefore 

<t> ~ ejr. 

With increasing distance from the charge, the potential decreases 
inversely as the distance. 
If the field is due not to one but to several charges e lt e 2 , . . ., it 



50 FIELDS [II 

follows from the principle of superposition that the potential at 
any point in space is given by the formula 

£ = -+-+•••, 

where r, is the distance of the point considered from the charge e t . 
When a charge e moves from a point where the potential is fa 
to a point where it is fa, the work done by the field forces is 
equal to the product of the charge and the difference of potential 
between the initial and final points: 

A 12 = e(fa-fa). 

Points at which the potential has a given value lie on a certain 
surface called an equipotential surface. When a charge moves on 
an equipotential surface, the work done by the field forces is 
zero. If the work is zero, the force must be perpendicular to the 
displacement. We can therefore say that the electric field at any 
point is perpendicular to the equipotential surface through that 
point. In other words, the lines of force are perpendicular to 
the equipotential surfaces. For example, for a point charge the 
lines of force are straight lines passing through the charge, and 
the equipotential surfaces are concentric spheres with the charge 
as centre. 

The electric potential has dimensions 

M = [U]l[e] 

= g^.cm^.sec -1 . 

This is the unit of potential in the CGSE system. In the SI 
system a unit 300 times smaller is used, called the volt : 

1 V = 1/300 CGSE unit of potential. 

If a charge of one coulomb moves between two points whose 
potentials differ by one volt, then the work done by the field 
forces is 3 X 10 9 x 1/300 = 10 7 erg, or one joule: 

1C.V= U. 



§20] GAUSS' THEOREM 51 

§20. Gauss' theorem 

We shall now define the important concept of electric flux. To 
explain this in terms of an analogy, let us imagine the space 
occupied by an electric field to be filled with some imaginary 
fluid whose velocity at every point is equal to the electric field 
in magnitude and direction. The volume of fluid passing through 
any surface per unit time is equal to the electric flux through that 
surface. 

The electric flux through a spherical surface of radius r due 
to a point charge e at its centre may be found as follows. The 
field in this case is, by Coulomb's law, E = e/r 2 . The velocity 
of the imaginary fluid is therefore also e/r 2 , and its flux is equal 
to this velocity multiplied by the area of the sphere, Airr 2 . Thus 
the flux is 

E . Airr 2 = Aire. 

We see that the flux is independent of the radius of the sphere 
and is determined only by the charge. It may be shown that, if 
the sphere is replaced by any other closed surface surrounding 
the charge, the electric flux through it is unchanged and is again 
equal to Aire. It should be emphasised that this important result 
is specifically a consequence of the fact that Coulomb's law 
involves the inverse square of the distance. 

Let us now consider the electric flux due not to one but to 
several charges. This may be determined by using the super- 
position property of the electric field. The flux through any 
closed surface is obviously equal to the sum of the fluxes from 
the individual charges within that surface. Since each such flux 
is equal to 4tt times the charge, the total electric flux through a 
closed surface is equal to 4tt times the algebraic sum of the 
charges within the surface. This is called Gauss' theorem. 




Fig. 20. 



52 FIELDS [II 

If there is no charge within the surface or the total charge 
within it is zero, the total electric flux through the surface is zero. 

Let us consider a narrow bundle of lines of force bounded by a 
surface itself consisting of lines of force (Fig. 20), and cut this 
bundle or tube of force by two equipotential surfaces 1 and 2; 
and let us determine the flux through the closed surface formed 
by the lateral surface of the tube of force and the equipotential 
surfaces 1 and 2. If there is no charge within this closed surface, 
the total flux through it will be zero. But the flux through the 
lateral surface of the tube is obviously zero, and the fluxes 
through the surfaces 1 and 2 must therefore be equal. The bundle 
of lines of force may be visualised as a jet of liquid. 

Let the fields at the cross-sections 1 and 2 be E x and E 2 and 
the areas of these cross-sections be 5 X and 5 2 . Since the tube of 
force is assumed to be narrow, the fields E x and E 2 may be 
regarded as constant over the respective cross-sections. We can 
therefore write the equality of the fluxes through the surfaces 
1 and 2 as 

since the field is perpendicular to the equipotential surface, the 
flux is just the product of the field and the surface area. The 
number N x of lines of force passing through the cross-section 
Si is equal to the number N 2 passing through S 2 , and we can 
therefore write 

NjSiEt = N 2 jS 2 E 2 . 

The quantities n x = N X \S X and n 2 = NjS 2 are the numbers of 
lines of force per unit area of the surfaces 1 and 2, which are 
orthogonal to the lines of force. Thus we see that the density or 
concentration of the lines of force is proportional to the field: 

njn 2 = EjE 2 . 

The description of the field by means of the lines of force 
therefore not only indicates the direction of the field but also 
gives an idea of its magnitude: where the lines of force are close 
together the electric field is strong, and where they are far apart 
it is weak. 



§21] ELECTRIC FIELDS IN SIMPLE CASES 53 

§2 1 . Electric fields in simple cases 

In many cases Gauss' theorem enables us to find the field due 
to composite charged bodies if the charge distribution in them is 
sufficiently symmetrical. 

As a first example, let us determine the field of a symmetrically 
charged sphere. The field of such a sphere is along its radii and 
depends only on the distance from the centre of the sphere. The 
field outside the sphere is therefore easily calculated. To do so, 
let us find the flux through a spherical surface of radius r con- 
centric with the sphere. This flux is evidently 4nr 2 E. By Gauss' 
theorem, the flux is 4n e, where e is the charge on the sphere. 
Hence 4irr 2 E = Aire, or 

E=e/r 2 . 

Thus the field outside the sphere is the same as that of a point 
charge at the centre of the sphere and equal to the charge on the 
sphere. Accordingly, the potential is also the same as that of a 
point charge: 

</> = e\r. 

The field within the sphere depends on how the charges are 
distributed within the sphere. If all the charges are on the 
surface of the sphere, then the field within the sphere is zero. If 
the charge is distributed uniformly through the volume of the 
sphere with density p per unit volume, then the field within the 
sphere can be found by applying Gauss' theorem to a spherical 
surface of radius r lying within the sphere: 

E . 4nr 2 — 47re r , 

where e r is the charge within the spherical surface. This charge is 
equal to the product of the charge density and the volume of 
a sphere of radius r. e r = 4nr 3 pl3. Thus 

477r 2 E = 47r.47r J p/3, 
or 

E = 4irprl3. 



54 



FIELDS 



[II 




We see that the field within a sphere of uniform charge per unit 
volume is proportional to the distance from the centre, while 
the field outside the sphere is inversely proportional to the square 
of this distance. Figure 2 1 shows the field of such a sphere as a 
function of the distance from its centre (a denoting the radius 
of the sphere). 





Fig. 22. 

As a second example, let us determine the field of a charged 
straight wire with charges distributed uniformly along it. If the 
wire is assumed to be sufficiently long, the effect of its ends may 
be neglected, i.e. it may be regarded as infinitely long. It is evident 
from symmetry that the field due to such a wire can have no 
component in either direction along the wire (since the two 
directions are entirely equivalent), and must therefore be per- 
pendicular to the wire at every point. It is then easy to determine 
the field of the wire. Let us consider the flux through a closed 
surface of radius r and length / with its axis along the wire 
(Fig. 22). Since the field is perpendicular to the axis, the flux 



§21] ELECTRIC FIELDS IN SIMPLE CASES 55 

through the ends of the cylinder is zero. The total flux through 
this closed surface therefore reduces to the flux through the 
lateral surface of the cylinder, which is evidently E . 2ttt/. By 
Gauss' theorem, this flux is Aire, where e is the charge on a 
length / of the wire; if q denotes the charge per unit length of the 
wire, then e = ql. Thus we have 

ItvyIE = Aire = 4irql, 
whence 

E = 2q/r. 

We see that the field due to a uniformly charged wire is inversely 
proportional to the distance r from the wire. 

Let us determine the potential of this field. Since the field E is 
along the radius at every point, its radial component E r is the 
same as its magnitude E. By the general relation between field 
and potential we therefore have 

-d<f>l dr = E = 2q/r, 
whence 

<f> = —2q loger+ constant. 

We see that in this case the potential is a logarithmic function of 
the distance from the wire. The constant in this formula can not 
be determined by using the condition that the potential should 
vanish at infinity, since the above expression becomes infinite 
as r -> oo. This is a result of the assumption that the wire is of 
infinite length, and signifies that the formula derived above can 
be used only for distances r which are small in comparison with 
the actual length of the wire. 

We may also find the field of a uniformly charged infinite plane. 
It is evident from symmetry that the field is perpendicular to the 
plane and has equal values (but opposite directions) at equal 
distances on either side of the plane. 

Let us consider the flux through the closed surface of a 
rectangular parallelepiped (Fig. 23) bisected by the charged 



56 



FIELDS 



[II 




Fig. 23. 



plane and having two faces parallel to that plane (the part of the 
plane lying within the parallelepiped is hatched in the diagram). 
The only non-zero flux is through these faces. Gauss' theorem 
therefore gives 

2SE = Aire = 4tt5o-, 

where S is the area of the face and cr the charge per unit area of 
the plane (surface density of charge). Thus we have 

E = 2tt(t. 

We see that the field of an infinite plane is independent of the 
distance from it. In other words, a charged plane creates a 
uniform electric field on either side of it. The potential of a 
uniformly charged plane is a linear function of the distance x 
from it: 

(J) = — 2 7TO-JC + constant. 

§22. Gravitational field 

As well as electrical interaction, gravitational interaction plays 
an extremely important part in Nature. This interaction is a 
property of all bodies, whether they are electrically charged or 
neutral, and is determined only by the masses of the bodies. The 
gravitational interaction between all bodies is an attraction, the 
force of interaction being proportional to the product of the 
masses of the bodies. 

If the bodies may be regarded as particles, the force of gravita- 
tional interaction is found to be inversely proportional to the 
square of the distance between them and proportional to the 



§22] GRAVITATIONAL FIELD 57 

product of their masses. Denoting the masses of the bodies by 
m 2 and n\ and the distance between them by r, we may write 
the gravitational force between them as 

F = —Grtixm^lr 1 , 

where G is a universal coefficient of proportionality independent 
of the nature of the interacting bodies; the minus sign shows that 
the force F is always attractive. This formula is called Newton's 
law of gravitation. 

The quantity G is called the gravitational constant; it is 
evidently the force of attraction between two particles each of 
unit mass at unit distance apart. The dimensions of the gravita- 
tional constant in the CGS system are 

[G] = [F][rY/[m] 2 

= (g.cm.sec~ 2 )cm 2 /g 2 
= cm 3 /g.sec 2 



and its value is 



G = 6-67 x 1(T 8 cm 3 /g.sec 2 . 



The extremely small value of G shows that the force of 
gravitational interaction can become considerable only for very 
large masses. For this reason the gravitational interaction plays 
no part in the mechanics of atoms and molecules. With increasing 
mass the importance of the gravitational interaction increases, 
and the motion of bodies such as the Moon, the planets and the 
artificial satellites is entirely determined by gravitational forces. 

The mathematical formulation of Newton's law of gravitation 
for particles is similar to that of Coulomb's law for point charges. 
Both the gravitational and the electrical force are inversely 
proportional to the square of the distance, the mass in the 
gravitational interaction corresponding to the charge in the 
electrical interaction. However, whereas electrical forces may 
be either attractive or repulsive, the gravitational forces are 
always attractive. 

The proportionality coefficient in Coulomb's law has been put 
equal to unity by appropriate choice of the unit of charge. We 



58 FIELDS [II 

could obviously proceed similarly with Newton's law of gravita- 
tion: by putting the gravitational constant equal to unity we 
should define a certain unit of mass. This would clearly be a 
derived unit relative to the centimetre and the second, and its 
dimensions would be cm 3 /sec 2 . The new unit of mass would be 
such as to impart an acceleration of 1 cm/sec 2 to an equal mass 
at a distance of 1 cm. Denoting this mass by /jl, we can write 

G = 6-67 X 1(T 8 cm 3 /g.sec 2 
= 1 cm 3 //a.sec 2 , 

whence /jl = 1-5 X 10 7 g= 15 tons. This new unit is obviously 
inconvenient, and it is therefore not used, but we can see that in 
principle a system of units could be constructed in which the 
only arbitrary units would be those of length and time, and 
derived units could be constructed for all other quantities, 
including mass. This system of units is not used in practice, but 
the possibility of constructing it again shows the arbitrariness of 
the CGS system. 

From the expression for the force of gravitational interaction 
between two particles we can easily find their potential energy 
U. Using the general relation between U and F: 

—dUjdr = F = —Gm 1 rrhlr l , 
we find 

U = —Gm 1 m 2 lr; 

the arbitrary constant in U is taken as zero so that the potential 
energy should vanish when the distance between the particles is 
infinite. This formula is similar to the formula 

U = e^Jr 

for the potential energy of the electrical interaction. 

We have given above the formulae for the force and potential 
energy of the gravitational interaction between two particles, 
but the same formulae are valid for the gravitational forces 
between any two bodies, provided that the distance between 
them is large compared with their size. For spherical bodies the 



§22] GRAVITATIONAL FIELD 59 

formulae are valid whatever the distance between them, r in 
this case denoting the distance between the centres of the spheres. 
The fact that the gravitational force acting on a particle is 
proportional to its mass enables us to define the gravitational 
field in the same way as the electric field. The force F acting on 
a particle of mass m is written 

F = mg, 

where the field g depends only on the masses and positions of 
the bodies which create the field. 

Since the gravitational field obeys Newton's law, which is 
mathematically similar to Coulomb's law for the electric field, 
Gauss' theorem is valid for the gravitational field also. The 
only difference is that the charge in Gauss' theorem is now 
replaced by the mass times the gravitational constant. Thus 
the gravitational flux through a closed surface is —AnrnG, where 
m is the total mass within the surface; the minus sign is due to 
the fact that gravitational forces are attractive. 

By using this theorem we can, for example, determine the 
gravitational field within a uniform sphere. This problem is 
identical with that of a uniformly charged sphere, discussed 
in §21. From the result obtained there we can write down 
immediately 

g = -47TGpr/3, 

where p is now the mass density of the sphere. 

The gravitational force acting on a body near the Earth's 
surface is called the weight P of the body. The distance of such 
a body from the centre of the Earth is R + z, where R is the 
Earth's radius and z the altitude of the body above the surface 
of the Earth. If the altitude z is very small compared with R it 
may be neglected, and the weight of the body is then 

P= GmM/R 2 , 

where M is the mass of the Earth. If this formula is written 

P= mg, 
then 

g = GM/R 2 . 



60 FIELDS [II 

The constant g is then called the acceleration due to gravity. It 
is the acceleration of free fall of a body in the Earth's gravitational 
field. 

At altitudes z such that the force of gravity may be regarded as 
constant, the potential energy of a body is given by the formula 

U = Pz = mgz. 

This can be seen from the general formula derived in §10 for the 
potential energy in a uniform field, if we use also the fact that in 
the present case the force is downwards, i.e. in the direction of 
decreasing z. 

In reality, the acceleration due to gravity, g, is not the same at 
different points on the Earth's surface, since the latter is not 
perfectly spherical. It should also be remembered that the 
rotation of the Earth about its axis causes a centrifugal force 
opposing the force of gravitation. It is therefore necessary to 
define an effective acceleration due to gravity, which is less than 
that on a hypothetical non-rotating Earth. At the poles this 
acceleration is g = 983-2 cm/sec 2 , and at the equator it is 
g = 978-0 cm/sec 2 . 

The value of g sometimes appears in the definition of the units 
of measurement of physical quantities (e.g. force and work). 
For this purpose a standard value is arbitrarily defined, 

g = 980-665 cm/sec 2 , 
which is very close to the value at latitude 45°. 

§2 3 . The principle of equivalence 

The fact that the force of gravity is proportional to the mass of 
the particle on which it acts (F = rag) is of very deep physical 
significance. 

Since the acceleration acquired by a particle is equal to the 
force acting on it divided by the mass, the acceleration w of a 
particle in a gravitational field is equal to the field itself, 

w= g, 

and is independent of the mass of the particle. In other words, 



§23] THE PRINCIPLE OF EQUIVALENCE 61 

the gravitational field has the remarkable property that all bodies, 
of whatever mass, are equally accelerated by it. This property 
was first discovered by Galileo in his experiments on the fall of 
bodies under the Earth's gravity. 

A similar behaviour of bodies would be found in a space where 
no external forces act on the bodies, if their motion were observed 
in a non-inertial frame of reference. Let us imagine, for example, 
a rocket in free motion in interstellar space, where the action of 
gravitational forces may be neglected. Objects within such a 
rocket will "float", remaining at rest relative to it. If the rocket 
is given an acceleration w, however, all the objects in it will "fall" 
to the floor with acceleration -w. This is the same as would be 
observed for a rocket moving without acceleration but subject 
to a uniform gravitational field -w towards the floor. It would not 
be possible to distinguish by experiment whether the rocket is 
moving with an acceleration or is in a uniform gravitational field. 

This similarity between the behaviour of bodies in a gravita- 
tional field and in a non-inertial frame of reference constitutes 
what is called the principle of equivalence. The fundamental 
significance of the similarity is fully shown in the theory of 
gravitation based on the theory of relativity . 

In the above discussion we have considered a rocket moving 
in space in the absence of a gravitational field. The same argument 
can be "inverted" by considering a rocket moving in a gravita- 
tional field, such as that of the Earth. A rocket moving "freely" 
(i.e. without engines) in such a field will receive an acceleration 
equal to the field g. The rocket is then a non-inertial frame of 
reference, and the effect of this on the motion relative to the 
rocket of the bodies within it is just balanced by the effect of the 
gravitational field. This brings about a state of "weightlessness"; 
that is, objects in the rocket behave as they would in an inertia! 
frame of reference in the absence of any gravitational field. Thus, 
by considering the motion relative to an appropriately chosen 
non-inertial frame of reference (in this case, the accelerated 
rocket), we can as it were "eliminate" the gravitational field. That 
is, of course, another aspect of the same principle of equivalence. 
The gravitational field which "appears" in an accelerated 
rocket is uniform throughout the rocket and is everywhere equal 
to -w. Actual gravitational fields, on the other hand, are never 
uniform. Thus the "elimination" of an actual gravitational field 



62 FIELDS [II 

by changing to a non-inertial frame of reference is possible only 
within small regions of space, over which the field changes so 
little that it may be regarded as uniform with sufficient accuracy. 
In this sense we may say that the gravitational field and the 
non-inertial frame of reference are only "locally" equivalent. 

§24. Keplerian motion 

Let us consider the motion of two bodies which attract each 
other in accordance with the universal law of gravitation, and 
first suppose that the mass M of one of the bodies is much 
greater than the mass m of the other body. If the distance r 
between the bodies is large in comparison with their size, we 
have a problem of the motion of a particle m in a central gravita- 
tional field due to a body M which may be regarded as at rest. 

The simplest motion in such a field is uniform motion in a 
circle round the centre of the field, i.e. round the centre of M. 
The acceleration is then towards the centre of the circle and is, 
as we know, equal to v?lr, where v is the velocity of the particle 
m. When multiplied by the mass m, this must equal the force 
exerted on the particle by the body M, i.e. 

mv 2 lr = GmM/r 2 , 

whence 

v = V(GM/r). 

Using this formula we can, in particular, determine the velocity 
of an artificial satellite moving near the Earth's surface. Replac- 
ing r by the Earth's radius R and GMlR 2 by the acceleration due 
due to gravity g, we obtain as the velocity of the satellite 

Vl = V(GMIR) = V(gR), 

called the first cosmic velocity. Substituting g ~ 980 cm/sec 2 , 
R = 6500 km, we find v t = 8 km/sec. 

The above formula for v gives a relation between the radius 
r of the orbit and the period T of one revolution. Putting 

v = 2irrlT, 



§ 24 1 KEPLERIAN MOTION 63 

we find 

r 

J 2 = 4 77 -2 A .3/ GM 

We see that the squares of the periods of revolution are pro- 
portional to the cubes of the orbit radii. This is Kepler's third law, 
named after the astronomer who in the early seventeenth century 
discovered empirically from observations of the planets the 
fundamental laws of the motion of two bodies under gravitational 
interaction (called Keplerian motion). These laws (of which the 
second, stating the constancy of the sectorial velocity for motion 
in a central field, has been discussed in §16) played an important 
part in Newton's discovery of the universal law of gravitation. 
Let us now determine the energy of the particle m. Its potential 
energy is, as we know, 

U = —GmM/r. 

Adding to this the kinetic energy \mv\ we find the total energy of 
the particle: 

E = \mv 2 - GmM/r, 

which is constant in time. 
For motion in a circle we have 

my 2 = GmM/r, 
and therefore 

E = -imv 2 = -GmMllr. 

We see that for motion in a circle the total energy of the particle 
is negative. This is in agreement with the results of §13, according 
to which, if the potential energy at infinity is zero, the motion will 
be finite for E < and infinite for E ^ 0. 

We have discussed a simple circular motion occurring under 
the action of an attractive force 

F = -GmM/r 2 . 



64 fields [n 

In such a field, however, the particle may move not only in a 
circle but also in an ellipse, hyperbola or parabola. For any of 
these conic sections one focus (in a parabola, the focus) is at the 
centre of force (Kepler's first law). Elliptical orbits evidently 
correspond to negative values of the total energy of the particle, 
E < (since the motion is finite). Hyperbolic orbits, with 
branches which go to infinity, correspond to positive values of 
the total energy, E > 0. Finally, for motion in a parabola E = 0. 
This means that in parabolic motion the velocity of the particle 
at infinity is zero. 

Using the formula for the total energy of the particle, we can 
easily find the minimum velocity which a satellite must have in 
order to move in a parabolic orbit, i.e. to escape from the Earth's 
attraction. Putting r = R in the formula 

E = \rriv 2 - GmMlr 

and equating E to zero, we obtain 

v 2 = V(2GMlR) = V(2gR), 

called the second cosmic velocity or velocity of escape. A 
comparison with the formula for the first cosmic velocity shows 

that 

y 2 = V2u! = 1 1-2 km/sec. 

Let us now see how the parameters of elliptical orbits are 
defined. The radius of a circular orbit may be expressed in 
terms of the energy of the particle: 

r = a/2\E\, 

where a = GmM. When the particle moves in an ellipse, the same 
formula gives the major semiaxis a of the ellipse: 

a = a/2\E\. 

The minor semiaxis b of the ellipse depends not only on the 
energy but also on the angular momentum L: 

b = LI\/(2m\E\). 



§24] KEPLERIAN MOTION 65 

The smaller the angular momentum L, the greater the elongation 
of the ellipse (for a given energy). 

The period of revolution in an ellipse depends only on the 
energy, and is given in terms of the major semiaxis by 

T 2 = 4TT 2 ma 3 /a. 

So far we have considered the case where the mass M of one 
of the bodies is much greater than the mass m of the other body, 
and we have therefore regarded the body M as being at rest. In 
reality, of course, both bodies are in motion, and they describe, 




Fig. 24. 

in the centre-of-mass frame, geometrically similar paths in the 
form of conic sections with a common focus at the centre of 
mass. Fig. 24 shows geometrically similar elliptical orbits of 
this kind. The particles m and M are at every instant at the ends 
of a line through the common focus O, and their distances from 
O are inversely proportional to their masses. 



CHAPTER III 

MOTION OF A RIGID BODY 

§2 5 . Types of motion of a rigid body 

So far we have considered the motion of bodies which might 
be regarded as particles under certain conditions. Let us now go 
on to consider motions in which the finite size of bodies is im- 
portant. Such bodies will be assumed to be rigid. In mechanics, 
this term means that the relative position of the parts of a body 
remains unchanged during the motion. The body thus moves as 
a whole. 

The simplest motion of a rigid body is one in which it moves 
parallel to itself; this is called translation. For example, if a 
compass is moved smoothly in a horizontal plane, the needle 
will retain a steady north-south direction and will execute a 
translational motion. 

In translational motion of a rigid body, every point in it has 
the same velocity and describes a path of the same shape, there 
being merely a displacement between the paths. 

Another simple type of motion of a rigid body is rotation about 
an axis. In rotation, the various points in the body describe 
circles in planes perpendicular to the axis of rotation. If in a 
time dt the body rotates through an angle d<f>, the path ds traversed 
in that time by any point P of the body is clearly ds = rd(f>, where 
r is the distance of P from the axis of rotation. Dividing by 
dt, we obtain the velocity of P: 

v = r dfyldt. 

The quantity d(f>ldt is the same at every point of the body and 
is the angular displacement of the body per unit time. It is called 
the angular velocity of the body, and we shall denote it by CI. 

Thus the velocities at various points in a rigid body rotating 
about an axis are given by 

v = rd, 

66 



§25] TYPES OF MOTION OF A RIGID BODY 67 

where r is the distance of the point from the axis of rotation, the 
velocity being proportional to this distance. 

The quantity Cl in general varies with time. If the rotation is 
uniform, i.e. the angular velocity is constant, 12 can be determined 
from the period of rotation T: 

il = 2tt/T. 

A rotation is denned by the direction of the axis of rotation and 
the magnitude of the angular velocity. These may be combined 
by means of the angular- velocity vector il, whose direction is 
that of the axis of rotation and whose magnitude is equal to the 
angular velocity. Of the two directions of the axis of rotation it 
is customary to assign to the angular-velocity vector the one 
which is related to the direction of rotation by the "corkscrew 
rule", i.e. the direction of motion of a right-handed screw rotating 
with the body. 





Fig. 25. 



These simple forms of motion of a rigid body are especially 
important because any motion of a rigid body is a combination 
of translation and rotation. This may be illustrated by the 
example of a body moving parallel to a certain plane. Let us 
consider two successive positions of the body, A x and A 2 (Fig. 
25). The body may evidently be brought from A x to A 2 in the 
following way. We first move the body by a translation from 
At to a position A' such that some point O of the body reaches 
its final position. If we then rotate the body about O through a 
certain angle </>, it will reach its final position A 2 . 

We see that the complete movement of the body consists of a 
translation from A x to A' and a rotation about O which finally 
brings the body to the position A 2 . The point O is clearly an 
arbitrary one: we could equally well carry out a translation of 
the body from the position A x to a position A" in which some 
other point O' , instead of O, has its final position, followed by 



68 MOTION OF A RIGID BODY [ill 

a rotation about O' which brings the body into its final position 
A 2 . It is important to note that the angle of this rotation is exactly 
the same as in the rotation about O, but the distance traversed in 
the translational motion of the points O and O' is in general 
different. 

The foregoing example shows (what is in fact a general rule) 
that an arbitrary motion of a rigid body can be represented as a 
combination of a translational motion of the whole body at the 
velocity of a point O in it and a rotation about an axis through 
that point. The translational velocity, which we denote by V, 
depends on the point in the body which is chosen, but the 
angular velocity ft does not depend on this choice: whatever 
the choice of the point O, the axis of rotation passing through it 
will have the same direction and the angular-velocity magnitude 
ft will be the same. In this sense we can say that the angular 
velocity ft is "absolute" and speak of the angular velocity of 
rotation of a rigid body without specifying the point through 
which the axis of rotation passes. The translational velocity is 
not "absolute" in this way. 

The "base" point O is usually taken to be the centre of mass of 
the body. The translational velocity V is then the velocity of the 
centre of mass. The advantages of this choice will be explained 
in §26. 

Each of the vectors V and ft is specified by its three components 
(in some system of coordinates). It is therefore necessary to 
specify only six independent quantities in order to know the 
velocity at any point in a rigid body. For this reason a rigid body 
is said to be a mechanical system with six degrees of freedom. 

§26. The energy of a rigid body in motion 

The kinetic energy of a rigid body in translational motion is 
very easily found. Since every point in the body is then moving 
with the same velocity, the kinetic energy is simply 

where V is the velocity of the body and M its total mass. This 
expression is the same as for a particle of mass M moving with 
velocity V. It is clear that translational motion of a rigid body 
is not essentially different from the motion of a particle. 



§26] THE ENERGY OF A RIGID BODY IN MOTION 69 

Let us now determine the kinetic energy of a rotating body. To 
do so, we imagine it divided into parts so small that they may be 
regarded as moving like particles. If m t is the mass of the ith 
part and r< its distance from the axis of rotation, then its velocity 
is v t = r t n, where ft is the angular velocity of rotation of the body. 
The kinetic energy is then hm^ 2 , and summation gives the total 
kinetic energy of the body: 

£km = ?m 1 v 1 2 + im 2 v 2 2 -\ 

= ift 2 (m 1 r 1 2 + m 2 r 2 2 +---). 

The sum in the parentheses depends on the rigid body concerned 
(its size, shape and mass distribution) and on the position of the 
axis of rotation. This quantity characteristic of a given solid 
body and a given axis of rotation is called the moment of inertia 
of the body about that axis, and is denoted by /: 

/ = m x rf + m 2 r 2 2 + • • •. 

If the body is continuous it must be divided into an infinite 
number of infinitesimal parts; the summation in the above formula 
is then replaced by integration. For example, the moment of 
inertia of a solid sphere of mass M and radius R, about an axis 
through its centre, is 2MR 2 I5; that of a thin rod of length / about 
an axis perpendicular to it through its midpoint is I = Ml 2 / 12. 

Thus the kinetic energy of a rotating body may be written as 

^kin = 2 /ft • 

This expression is formally similar to that for the energy of 
translation, but the velocity V is replaced by the angular velocity 
ft, and the mass by the moment of inertia. This is one example 
showing that the moment of inertia in rotation corresponds to 
the mass in translation. 

The kinetic energy of a rigid body moving in an arbitrary 
manner can be written as the sum of the translational and 
rotational energies if the point O in the method of separating 
the two motions, described in §25, is taken to be the centre of 
mass of the body. Then the rotational motion will be a motion of 



70 MOTION OF A RIGID BODY [ill 

the points in the body about its centre of mass, and there is an 
exact analogy with the separation of the motion of a system of 
particles into the motion of the system as a whole and the 
"internal" motion of the particles relative to the centre of mass 
(§12). We saw in §12 that the kinetic energy of the system also 
falls into two corresponding parts. The "internal" motion is here 
represented by the rotation of the body about the centre of mass. 
The kinetic energy of a body moving in an arbitrary manner is 
therefore 

The suffix signifies that the moment of inertia is taken about an 
axis through the centre of mass. 

[It should be noted, however, that in this form the result is of 
practical significance only if the axis of rotation has a constant 
direction in the body during the motion. Otherwise the moment of 
inertia has to be taken about different axes at different times, 
and is therefore no longer a constant.] 

Let us consider a rigid body rotating about an axis Z which 
does not pass through the centre of mass. The kinetic energy 
of this motion is E kin = ?Iil 2 , where / is the moment of inertia 
about the axis Z. On the other hand, we may regard this motion 
as consisting of a translational motion with the velocity V of 
the centre of mass and a rotation (with the same angular velocity 
il) about an axis through the centre of mass parallel to the axis Z. 
If the distance of the centre of mass from the axis Z is a, then 
its velocity V = ail. The kinetic energy of the body may therefore 
be written also as 

E kin = iMV 2 + iI il 2 
= KMa 2 + I )il 2 . 

Hence 

I = I + Ma 2 . 

This formula relates the moment of inertia of the body about 
any axis to its moment of inertia about a parallel axis through 
the centre of mass. It is evident that / is always greater than 



§26] 



THE ENERGY OF A RIGID BODY IN MOTION 



71 



7 . In other words, for a given direction of the axis the minimum 
value of the moment of inertia is reached when the axis passes 
through the centre of mass. 




V////////////////////////////// 
Fig. 26. 

If a rigid body moves under gravity, its total energy E is the 
sum of the kinetic and potential energies. As an example, let 
us consider the motion of a sphere on an inclined plane (Fig. 
26). The potential energy of the sphere is Mgz, where M is the 
mass of the sphere and z the height of its centre. The law of 
conservation of energy therefore gives 

E = \MV 2 + il [l 2 + Mgz = constant. 

Let us suppose that the sphere rolls without slipping. Then 
the velocity v of its point of contact with the plane is zero. On 
the other hand, this velocity consists of the velocity V of the 
translational motion of the point down the plane (together with 
the whole sphere) and the velocity of the point in the opposite 
direction (up the plane) in its rotation about the centre of the 
sphere. The latter velocity is ilR, where R is the radius of the 
sphere. The equation v = V— flR = thus gives 

n = VIR. 

Substituting this expression in the law of conservation of 
energy and assuming that at the initial instant the velocity of 
the sphere is zero, we find the velocity of the centre of mass of 
the sphere when it has descended a vertical distance h: 



V = 



2gk 



;)• 



A + h/MR 2 / 
This is, as we should expect, less than the velocity of free fall 



72 



MOTION OF A RIGID BODY 



[III 



of a particle or of a non-rotating body (from the same height h), 
since the decrease Mgh of the potential energy goes not only to 
increase the kinetic energy of the translational motion but also 
to increase that of the rotation of the sphere. 

§27. Rotational angular momentum 

In rotational motion of a body its angular momentum plays a 
part similar to that of the momentum in the motion of a particle. 
In the simple case of a body rotating about a fixed axis Z, this 
part is played by the angular-momentum component along that 
axis. 




To calculate this component, we divide the body into elemen- 
tary parts, as in calculating the kinetic energy. The angular 
momentum of the ith element is m^ X y it where R t is the radius 
vector of this element relative to some point O on the axis Z 
about which the angular momentum is to be determined (Fig. 27). 
Since every point in the body moves in a circle round the axis 
of rotation, the velocity v t is tangential to the circle in Fig. 27, 
i.e. is in a plane at right angles to OZ. We can resolve the vector 
Ri into two vectors, one along the axis and the other r t per- 
pendicular to the axis. Then the product m^ X \ t is just the part 
of the angular momentum which is parallel to the axis Z (it will 
be recalled that the vector product of two vectors is perpendicular 
to the plane through those vectors). Since the vectors r t and v t 



§28] THE EQUATION OF MOTION OF A ROTATING BODY 73 

are mutually perpendicular (being a radius of the circle and a 
tangent to it), the magnitude of the product r, X \ t is just r^, 
where r, is the distance of the element m { from the axis of rotation. 
Finally, since v t = Or*, we conclude that the component of the 
angular momentum of the element ra* along the axis of rotation 
is mtrfCl. The sum mrfil + w 2 r 2 2 flH gives the required com- 
ponent L z of the total angular momentum of the body along the 
axis Z. This quantity is also called the angular momentum of the 
body about that axis. 

When the common factor H is removed from the above sum 
there remains a sum which is just the expression for the moment 
of inertia /. Thus we have finally 

L z = m, 

i.e. the angular momentum of the body is equal to the angular 
velocity multiplied by the moment of inertia of the body about the 
axis of rotation. The analogy between this expression and the 
expression m\ for the momentum of a particle should be noticed: 
the velocity v is replaced by the angular velocity and the mass 
is again replaced by the moment of inertia. 

If no external forces act on the body, its angular momentum 
remains constant: it rotates "by inertia" with a constant angular 
velocity H. Here the constancy of H follows from that of L z 
because we have assumed that the body itself is unchanged during 
the rotation, i.e. its moment of inertia is unchanged. If the relative 
position of the parts of the body, and therefore its moment of 
inertia, vary, then in free rotation the angular velocity will also 
vary in such a way that the product /O remains constant. For 
example, if a man holding weights in his hands stands on a 
platform rotating with little friction, by extending his arms he 
increases his moment of inertia, and the conservation of the 
product /O causes his angular velocity of rotation to decrease. 

§2 8 . The equation of motion of a rotating body 

The equation of motion of a particle gives, as we know, the 
relation between the rate of change of its momentum and the 
force acting on it (§7). The translational motion of a rigid body 
is essentially the same as the motion of a particle, and the equation 
of this motion consists of the same relation between the total 



74 MOTION OF A RIGID BODY [ill 

linear momentum P = MV of the body and the total force F 
acting on it: 

dVldt = MdY/dt = F. 

In rotational motion there is a corresponding equation relating 
the rate of change of the angular momentum of the body to the 
torque acting on it. Let us find the form of this relation, again 
taking the simple case of rotation of the body about a fixed axis Z. 

The angular momentum of the body about the axis of rotation 
has already been determined. Let us now consider the forces 
acting on the body. It is clear that forces parallel to the axis of 
rotation can only move the body along that axis and can not 
cause it to rotate. We can therefore ignore such forces and 
consider only those in the plane perpendicular to the axis of 
rotation. 

The corresponding torque K z about the axis Z is given by the 
magnitude of the vector product rXF, where r is the vector 
giving the distance of the point of application of the force F from 
the axis. By the definition of the vector product we have 

K z = Fr sin 0, 

where 6 is the angle between F and r; in Fig. 28 the axis Z is at 
right angles to the plane of the diagram and passes through O, 
and A is the point of application of the force. We can also write 

K z = h F F, 

where h F = r sin is the moment arm of the force about the axis 
(the distance of the line of action of the force from the axis). 

According to the relation established in §15 between the rate 
of change of angular momentum and the applied torque, we can 
write the equation 

dLzldt = K z or IdMdt = K z . 

This is the equation of motion of a rotating body. The derivative 
d£lldt may be called the angular acceleration. We see that it is 
determined by the torque acting on the body, just as the accelera- 
tion of the translational motion is determined by the force acting. 



§28] THE EQUATION OF MOTION OF A ROTATING BODY 75 

If there are several forces acting on the body, then K z in the 
above equation must of course be taken as the sum of the torques. 
It must be remembered that K z is derived from a vector, and 
torques tending to turn the body in opposite directions about 
the axis must be given opposite signs. Those torques have 
positive signs which tend to rotate the body in the direction in 
which the angle of the rotation of the body about the axis is 
measured ($ is the angle whose time derivative is the angular 
velocity of rotation of the body: O, = d<f>/dt). 

We may also note that the point of application of a force in a 
rigid body may be displaced in any manner along its line of action 
without affecting the properties of the motion. This will evidently 
leave unchanged the arm of the force and therefore the torque. 
The condition for equilibrium of a body which can rotate about 
an axis is evidently that the sum of the torques acting on it should 
be zero. This is the law of torques (or law of moments). A 
particular case is the familiar lever rule which gives the condition 
of equilibrium for a rod able to rotate about one point in it. 



There is a simple relation between the torque acting on a body 
and the work done in a rotation of the body. The work done by 
a force F when the body rotates about the axis through an 
infinitesimal angle d<f> (Fig. 28) is equal to the product of the 
displacement ds = rd<f> of the point A where the force is applied 
and the component F s = F sin of the force in the direction of 
motion: 

F s ds ~ Fr sin d(f> = K z d4>. 

We see that the torque about the axis is equal to the work done 
per unit angular displacement. On the other hand, the work done 



76 MOTION OF A RIGID BODY [ill 

on the body is equal to the decrease in its potential energy. We 
can therefore put K z d<$> = —dU, or 

K z = -dUld<f>. 

Thus the torque is equal to minus the derivative of the potential 
energy with respect to the angle of rotation of the body about the 
given axis. The analogy between this relation and the formula 
F = —dUldx should be noticed; the latter relates the force to the 
change in potential energy in motion of a particle or translational 
motion of a body. 

It is easy to see that the equation of motion of a rotating body 
is, as it should be, in accordance with the law of conservation of 
energy. The total energy of the body is 

and its conservation is expressed by the equation 
j t (im 2 +U)=0. 

From the rule for differentiating a function of a function we have 

dU = dUd± = _ n 
dt dct> dt z ' 

The derivative dWIdt = 2ft dMdt. Substituting these expressions 
and cancelling the common factor ft, we again obtain the equation 
Idn/dt = K z . 

At the end of § 15 it has been mentioned that there is a relation 
between the law of conservation of the angular momentum of a 
closed system and the isotropy of space. To establish this rela- 
tion, we have to prove that the vanishing of the total torque acting 
in the system is a consequence of the fact that the properties of 
a closed system are unchanged by any rotation of it as a whole 
(that is, as if it were a rigid body). By applying the relation 
dUld<f> = —K z to the internal potential energy of the system, 
taking K z to be the total torque on all the particles, we see that the 
condition for the potential energy to be unchanged by a rotation 
of the closed system about any axis is in fact that the total torque 
should be zero. 



§29] RESULTANT FORCE 77 

§29. Resultant force 

If several forces act on a rigid body, the motion of the body 
depends only on the total force and the total torque. This some- 
times enables us to replace the forces acting on the body by a 
single force called the resultant. It is evident that the magnitude 
and direction of the resultant are given by the vector sum of the 
forces, and its point of application must be so chosen that the 
resultant torque is equal to the sum of the torques. 

The most important such case is that of the addition of parallel 
forces, which includes, in particular, the addition of the forces of 
gravity acting on the various parts of a rigid body. 




Let us consider a body and determine the total torque of gravity 
about an arbitrary horizontal axis (Z in Fig. 29). The force of 
gravity acting on the element m t of the body is m^, and its 
moment arm is the coordinate x t of the element. The total torque 
is therefore 

Kz = m l gx 1 + m 2 gx 2 H . 

The magnitude of the resultant is equal to the total weight 
{m x + 1712 + ■ ■ )g of the body, and if the coordinate of its point of 
application is denoted by X, the torque K z has the form 

Kz= (m 1 + m 2 +- • -)gX. 

Equating these two expressions, we find 

X= (m 1 x l + m 2 x 2 -\ )l(m 1 + m 2 -\ ). 



78 MOTION OF A RIGID BODY [ill 

This is just the jc coordinate of the centre of mass of the body. 

Thus we see that all the forces of gravity acting on the body 
can be replaced by a single force equal to the total weight of the 
body and acting at its centre of mass. For this reason the centre 
of mass of the body is often called its centre of gravity. 

The reduction of a system of parallel forces to a single resultant 
force is not possible, however, if the sum of the forces is zero. 
The effect of such forces can be reduced to that of a couple, 
i.e. two forces equal in magnitude and opposite in direction. It 
is easily seen that the sum K z of their torques about any axis Z 
perpendicular to the plane of action of two such forces is equal 
to the product of either force F and the distance h between their 
lines of action (the arm of the couple): 

K z = Fh. 

The effect of the couple on the motion of the body depends only 
on this quantity, called the moment of the couple. 

§30. The gyroscope 

In §27 we have derived the component L z of the angular 
momentum of a body along the axis of rotation. For a body 
rotating about a fixed axis, only this component of the vector 
L is important. The simple relation between this component and 
the angular velocity of rotation ft (Lz = /ft) has the result that 
the entire motion is simple. 

If the axis of rotation is not fixed, however, it is necessary to 
consider the entire vector L as a function of the angular-velocity 
vector ft. This function is more complicated: the components 
of the vector L are linear functions of those of ft, but the direc- 
tions of the two vectors are in general different. This considerably 
complicates the nature of the motion of the body in the general 
case. 

Here we shall consider only one example of the motion of a 
body with a freely variable axis of rotation, namely the gyroscope; 
this is an axially symmetric body rotating rapidly about its 
geometrical axis. 

In such a rotation the angular momentum L is along the axis of 
the body, like the angular- velocity vector ft. This is obvious 
simply from considerations of symmetry: since the motion is 



THE GYROSCOPE 



79 



§30] 

axially symmetric, there is no other preferred direction which 
could be taken by the vector L. 

So long as no external forces act on the gyroscope, its axis will 
remain in a fixed direction in space, since by the law of conserva- 
tion of angular momentum the direction (and the magnitude) of 
the vector L does not vary. If external forces are applied to the 
gyroscope, its axis will begin to deviate. It is this movement of 
the gyroscope axis (called precession) which we shall discuss. 

The change in direction of the gyroscope axis consists in a 
rotation about some other axis, so that the total angular- velocity 
vector is not along the geometrical axis of the body. The angular- 
momentum vector L likewise will not coincide with this axis (nor 
with the direction of ft). But if the primary rotation of the 
gyroscope is sufficiently rapid, and the external forces are not 
too great, the rate of rotation of the gyroscope axis will be 
relatively small, and the vector ft (and therefore L) will always 
be in a direction close to the axis. Hence, if we know how the 
vector L varies, we know approximately how the gyroscope axis 
moves. The change in angular momentum is given by the equation 



dL/dt = K, 



where K is the applied torque. 




Fig. 30. 



For example, let forces F acting in the yz plane be applied at 
the ends of the gyroscope axis (the z axis in Fig. 30). Then the 
moment K of the couple is along the jc axis, and the derivative 



80 



MOTION OF A RIGID BODY 



[III 



dLldt will also be in that direction. Hence the angular momentum 
L, and therefore the gyroscope axis, will deviate in the direction 
ofthejcaxis. 

Thus the application of a force to the gyroscope causes its axis 
to turn in a direction perpendicular to the force. 




ZT7 



Fig. 31. 



An example of a gyroscope is a top supported at its lowest 
point. [In the following discussion we neglect friction at the 
support.] The top is subject to the force of gravity, whose direc- 
tion is constant, namely vertically downwards. This force is 
equal to the weight of the top: P = Mg, where M is the mass of 
the top, and acts at its centre of gravity (C in Fig. 3 1). The torque 
about the point of support O is in magnitude K = PI sin (where / 
is the distance OC and 6 the angle between the axis of the top 
and the vertical), and its direction is always perpendicular to the 
vertical plane through the axis of the top. Under the action of 
this torque the vector L (and therefore the axis of the top) will 
be deflected, remaining constant in magnitude and at a constant 
angle 6 to the vertical, i.e. describing a cone about the vertical. 

It is easy to determine the angular velocity of precession of the 
top. We denote this by o> to distinguish it from that of the rotation 
of the top about its own axis, which is denoted by ft - 

In an infinitesimal time dt the vector L receives an increment 
dL = Kdt perpendicular to itself and lying in a horizontal plane 
(Fig. 31). 



§31] INERTIA FORCES 81 

Dividing this by the component of the vector L in this plane, 
we find the angle d<f> through which the component turns in the 
time dt: 

d<}> = . dt. 
L sin 

The derivative d<f>l dt is evidently the required angular velocity of 
precession. Thus 

o) = K/L sin 0. 

Substituting A: = Mgl sin and L = ICl (where / is the moment of 
inertia of the top about its axis), we obtain finally 

(o = Mglim . 

The rotation of the top, it will be remembered, has been 
assumed sufficiently rapid. We can now make this condition 
more precise: we must have Cl > co. 

Since 

w/n = M g i/m 2 , 

we see that this condition implies that the potential energy of 
the top in the gravitational field (Mgl cos 6) must be small in 
comparison with its kinetic energy (i/O 2 ). 

§31. Inertia forces 

So far we have considered the motion of bodies with respect to 
inertial frames of reference, and have discussed only in §23 a 
frame of reference in accelerated translational motion (an 
accelerated rocket). We have seen that, from the point of view 
of an observer moving with the rocket, the fact that the frame of 
reference is non-inertial is perceived through the appearance of 
a force field equivalent to a uniform field of gravity. 

The additional forces which appear in non-inertial frames of 
references are called inertia forces. Their characteristic feature 
is that they are proportional to the mass of the body on which 
they act. This makes them similar to gravitational forces. 



82 MOTION OF A RIGID BODY [ill 

Let us now consider how motion occurs with respect to a 
rotating frame of reference, and the inertia forces which appear. 
The Earth itself, for example, is such a frame of reference; owing 
to the Earth's daily rotation, the frame of reference in which the 
Earth is fixed is, strictly speaking, non-inertial, although the 
resulting inertia forces are comparatively small because of the 
slowness of the rotation. 

For simplicity, let us assume that the frame of reference is a 
disc rotating uniformly (with angular velocity fl) and consider a 
simple motion on it: that of a particle moving uniformly along 
the edge of the disc. Let the velocity of this particle relative to 
the disc be v n , the suffix n indicating that the frame of reference 
is non-inertial. The velocity v t of the particle relative to a fixed 
observer (inertial frame of reference) is evidently the sum of v n 
and the velocity of the points on the edge of the disc itself. The 
latter is Cir, where r is the radius of the disc. Hence 

v t = v n + Clr. 

It is easy to determine the acceleration w t of the particle in the 
inertial frame of reference. Since the particle moves uniformly in 
a circle of radius r with velocity v u we have 

w t = vflr 

= ivVr + 2ftt> B + ft 2 r. 

Multiplying this acceleration by the mass m of the particle, we 
find the force F acting on the particle in the inertial frame of 
reference: 



mwi 



Let us now consider how this motion will appear to an observer 
located on the disc and regarding it as being at rest. This observer 
also will see the particle moving uniformly in a circle of radius r, 
but with velocity v n . The acceleration of the particle relative to 
the disc is therefore 

w„ = v n 2 /r 



§31] INERTIA FORCES 83 

towards the centre of the disc. Regarding the disc as being at rest, 
the observer multiplies w n by the mass of the particle and takes 
this product to be the force F n acting on the particle: 

F n = mw n . 

Since 

w n = w i -inv n -Ct 2 r, 

and mwi = F, we find that 

F n = F — 2mClv n — mQPr. 

Thus we see that, relative to the rotating frame of reference, the 
particle is subject not only to the "true" force F but also to two 
additional forces, —mCl 2 r and —2mClv n . The former of these inertia 
forces is called the centrifugal force, and the latter the Coriolis 
force. The minus signs indicate that in this case both forces are 
directed away from the axis of rotation of the disc. 

The centrifugal force is independent of the velocity v n , i.e. it 
exists even if the particle is at rest relative to the disc. For a 
particle at a distance r from the axis of rotation of the frame of 
reference this force is always equal to mQ?r and is directed 
radially away from the axis. 

Having defined the centrifugal force, we may also define the 
centrifugal energy as the potential energy of a particle in the 
centrifugal force field. According to the general formula relating 
the force and the potential energy, we have 

—dU C fldr = m£l 2 r, 

whence 

U ct = —kmfflr 2 + constant. 

The arbitrary constant may reasonably be taken as zero, the 
potential energy thus being measured from its value on the axis 
of rotation (r = 0), where the centrifugal force is zero. 
The centrifugal force can reach very large values in specially 



84 MOTION OF A RIGID BODY [ill 

designed centrifuges. On the Earth, it is very small. Its maximum 
value occurs at the equator, where the force on a particle of 
mass 1 g is 

mWR = 1 x (2tt/24 X 60 X 60) 2 X 6-3 X 10 8 dyn 
= 3-3dyn 

(# = 6-3 X 10 8 cm being the Earth's radius). This force therefore 
decreases the weight of a body by 3-3 dyne per gram, i.e. by 
about 0-3%. 

The second inertia force, the Coriolis force, is quite different in 
type from any of the forces so far discussed. It acts only on a 
particle which is in motion (relative to the frame of reference 
considered) and depends on the velocity of that motion. It is, 
on the other hand, independent of the position of the particle 
relative to the frame of reference. In the example discussed above, 
its magnitude is 2mCtv n and its direction is away from the axis of 
rotation of the disc. It can be shown that in general the Coriolis 
inertia force on a particle moving with any velocity \ n relative to 
a frame of reference rotating with angular velocity 11 is 

2m\ n X ft. 

In other words, it is perpendicular to the axis of rotation and to 
the velocity of the particle, and its magnitude is 2mv n CL sin d, 
where is the angle between \ n and ft. When the direction of the 
velocity v w is reversed, so is that of the Coriolis force. 

Since the Coriolis force is always perpendicular to the direction 
of motion of the particle, it does no work, but simply changes the 
direction of motion of the particle without altering its velocity. 

Although the Coriolis force is usually very small on the Earth, 
it does bring about certain specific effects. Because of this force, 
a freely falling body will not move exactly vertically, but will be 
deflected slightly eastwards. The deviation is very slight, how- 
ever. For example, calculation shows that the deflection in a fall 
from a height of 100 m in latitude 60° is only about 1 cm. 

The Coriolis force accounts for the behaviour of the Foucault 
pendulum, which used to serve as a demonstration of the Earth's 
rotation. If there were no Coriolis force, the plane of oscillation 
of a pendulum swinging near the Earth's surface would remain 



§31] INERTIA FORCES 85 

fixed (relative to the Earth). The effect of this force is to cause 
the plane of oscillation to rotate round the vertical with angular 
velocity ft sin $, where ft is the angular velocity of the Earth's 
rotation and the latitude of the point at which the pendulum is 
suspended. 

The Coriolis force plays a large part in meteorological phenom- 
ena. For example, the trade winds, which blow from the tropics 
to the equator, would blow directly from the north in the northern 
hemisphere and from the south in the southern hemisphere, if 
the Earth were not rotating. The Coriolis force causes a westward 
deflection of these winds. 



CHAPTER IV 

OSCILLATIONS 



§32. Simple harmonic oscillations 

We have seen in § 1 3 that a one-dimensional motion of a particle 
in a potential well is periodic, i.e. is repeated at equal intervals of 
time. The interval after which the motion is repeated is called the 
period of the motion. If this is denoted by T, then the particle has 
the same position and velocity at times t and t + T. 

The reciprocal of the period is called the frequency, and will be 
denoted by v. 

v= 1IT; 

it gives the number of times per second that the motion is re- 
peated. Its dimensions are evidently 1/sec, and the unit of 
measurement of frequency, corresponding to a period of 1 sec, 
is called the hertz (Hz): 1 Hz = 1 sec -1 . 

There is obviously an infinite variety of types of periodic 
motion. The simplest periodic functions are the trigonometric 
sine and cosine functions, and the simplest periodic motion is 
therefore one in which the coordinate of the particle varies 
according to 

x = A cos (cot + a), 

where A, w and a are constants. Such a periodic motion is called 
a simple harmonic oscillation. 

The quantities A and co have a simple physical significance. 
Since the period of the cosine is 2ir, the period T of the motion is 
related to to by 

T — 2ttI(x). 



§32] SIMPLE HARMONIC OSCILLATIONS 87 

Hence we see that to differs from v by a factor 2ir: 

o) = 2ttv. 

The quantity a> is called the angular frequency; it is generally 
used in physics to describe oscillations, and is often called simply 
the frequency. 

Since the maximum value of the cosine is unity, the maximum 
value of the coordinate x is A. This maximum value is called the 
amplitude of the oscillation, and x varies from— A to A. 

The argument cot + a of the cosine is called the phase of the 
oscillation, and a is the initial phase (at time t = 0). 

The velocity of the particle is 

v = dx/dt = —Aco sin (cot + a). 

We see that the velocity also varies harmonically but the cosine 
is replaced by a sine. If this expression is written 

v = A (o cos (cot + a + jtt) 

we can say that the velocity "leads" the coordinate by hr in 
phase. The amplitude of the velocity is equal to the amplitude of 
the displacement multiplied by the frequency co. 

Let us now ascertain what the force acting on the particle must 
be in order to cause it to execute simple harmonic oscillations. 
To do so, we find the acceleration of the particle in such a motion. 
We have 

w = dvjdt = —Aco 2 cos (cot + a). 

This quantity varies in the same manner as the coordinate of the 
particle, but differs from it in phase by tt. Multiplying w by the 
mass m of the particle and noticing that A cos (cot + a) = x, we 
obtain the following expression for the force: 

F — —mco 2 x. 

Thus, in order that a particle should execute simple harmonic 
oscillations, the force acting on it must be proportional to the 



88 OSCILLATIONS [IV 

displacement of the particle and in the opposite direction. An 
elementary example is that of the force exerted on a body by a 
stretched (or compressed) spring; this is proportional to the 
elongation (or shortening) of the spring and is always in a direc- 
tion such that the spring tends to regain its original length. Such 
a force is often called a restoring force. 

The dependence of the force on the position of a particle in 
physical problems is very often found to be of this type. If a 
body is in a position of stable equilibrium (at the point x = 0, say) 
and is then moved slightly in either direction from this position, 
a force F results which tends to return the body to its equilibrium 
position. As a function of the position x of the body, the force 
F = .F(x) is represented by a curve passing through the origin: 
at the point jc = the force F = 0, and it has opposite signs on 
either side of this point. Over a short range of values of x, this 
curve can be approximated by a section of a straight line, so that 
the force is proportional to the displacement x. Thus, if the body 
undergoes a slight displacement from the equilibrium position 
and is then left to itself, its return to the equilibrium position will 
give rise to simple harmonic oscillations. 

Motions in which a body deviates only slightly from a position 
of equilibrium are called small oscillations. Thus small oscilla- 
tions are simple harmonic. The frequency of these oscillations is 
determined by the rigidity with which the body is fixed; this gives 
the relation between the force and the displacement. If the force 
is related to the displacement by 

F = —kx, 

where k is a coefficient called the stiffness, a comparison with the 
expression for the force in simple harmonic motion, F = —mo) 2 x, 
shows that the frequency of the oscillations is 

co = V(*/m). 

It must be emphasised that the frequency depends only on the 
properties of the oscillating system (the rigidity with which the 
body is fixed, and the mass of the body), and not on the amplitude 
of the oscillations. A given body executing oscillations of various 
amplitudes does so with the same frequency. This is a very 



§32] SIMPLE HARMONIC OSCILLATIONS 89 

important property of small oscillations. The amplitude, on the 
other hand, is determined not by the properties of the system 
itself but by the initial conditions of its motion, i.e. by the initial 
disturbance which causes the system to be no longer at rest. The 
oscillations of the system resulting from an initial disturbance, 
after which the system is left to itself, are called natural 
oscillations. 

The potential energy of an oscillating particle is easily found by 
noting that 

dU/dx = -F = kx, 

whence 

U = ?kx 2 + constant. 

Choosing the constant so that the potential energy is zero in 
the equilibrium position (x = 0), we have finally 

U = ikx 2 , 

i.e. the potential energy is proportional to the square of the 
displacement of the particle. 

Adding the potential energy to the kinetic energy, we find the 
total energy of the oscillating particle: 

E = \mv 2 + \kx 2 

= jmA 2 o) 2 sin 2 (cot + a) + ?mA 2 co 2 cos 2 (cot + a) 

or 



Thus the total energy is proportional to the square of the ampli- 
tude of the oscillations. It should be noted that the kinetic and 
potential energies vary as sin 2 (cot + a) and cos 2 (cot + a), so that 
when one increases the other decreases. In other words, the 
process of oscillation involves a periodic transfer of energy 
between potential and kinetic and vice versa. The mean values 
(over the period of the oscillation) of the potential and kinetic 
energies are each equal to \E. 



90 OSCILLATIONS [IV 

§33. The pendulum 

As an example of small oscillations, let us consider oscillations 
of a simple pendulum; this consists of a particle suspended by a 
string in the Earth's gravitational field. 

Let us deflect the pendulum from its equilibrium position 
through an angle <j> and determine the force then acting on it. 
The total force on the pendulum is mg, where m is the mass of 
the pendulum and g the acceleration due to gravity. We resolve 
this force into two components (Fig. 32), one along the string 
and the other perpendicular to it. The first component is balanced 
by the tension in the string, while the second component causes 
the motion of the pendulum. This component is evidently 

F = — mg sin </>. 

For small oscillations the angle (/> is small, and sin $ is approxi- 
mately equal to (f> itself, so that F ~ —mg(f>. Since l<f) is the 
distance x through which the particle moves (/ being the length of 
the pendulum), we can write 

F = —mgxjl. 

Thus we see that the stiffness k = mgjl for small oscillations of 
a pendulum. The frequency of these oscillations is therefore 

*> = Vte//). 

The period of the oscillations is 

r = 27r/a) = 27rV(//g). 

The length of a pendulum with period T = 1 sec, for the standard 
acceleration due to gravity given at the end of §22, is / = 24-84 cm. 
The manner in which the period of a pendulum depends on its 
length and the acceleration due to gravity can also be easily 
determined from dimensional considerations. The quantities 
available to characterise the mechanical system in question are 
m, I and g, with dimensions 

[m] = g, [/] = cm, [g] = cm/sec 2 . 



§33] 



THE PENDULUM 



91 




Fig. 32. 



The period T can depend only on these quantities. Since only m 
contains the dimension g, and the dimensions of the required 
quantity [T] = sec do not contain g, it is clear that T cannot 
depend on m. From the two remaining quantities / and g we can 
eliminate the dimension cm (which is not present in T) by taking 
the ratio II g. Finally, by taking the square root V(llg) we obtain 
a quantity having the dimensions sec, and it is clear from the 
foregoing argument that this is the only way in which such a 
quantity can be obtained. We can therefore assert that the period 
T must be proportional to V(//g); the numerical value of the 
coefficient of proportionality can not, of course, be determined 
by this method. 

So far we have discussed small oscillations in terms of a single 
particle, but the results obtained in fact apply also to the oscilla- 
tions of more complex systems. As an example, let us consider 
the oscillations of a rigid body that can rotate under gravity about 
a horizontal axis. This is called a compound pendulum. 

We have seen in §28 that the laws of motion of a rotating body 
are formally identical with those of a particle, the coordinate x 
being replaced by the angle of rotation </>, the mass by the moment 
of inertia / of the body about the axis of rotation, and the force F 
by the torque K z . 

In the present case the torque of gravity about the axis of 
rotation is K z = —mga sin </>, where m is the mass of the body, a 
the distance of its centre of gravity C from the axis of rotation 
(which passes through the point O at right angles to the plane of 
Fig. 33), and <f> the angle of deflection of the line OC from the 
vertical. The minus sign shows that the torque K z tends to 



92 



OSCILLATIONS 



[IV 




decrease the angle <)>. In small oscillations the angle </> is small, 
and therefore K z « - mga4>. Comparing this with the expression 
for the restoring force F = — kx in oscillations of a particle, we 
see that the stiffness coefficient k is now replaced by mga. Thus, 
by analogy with the formula <o = \ / (klm), we can write down 
the following expression for the frequency of oscillations of a 
compound pendulum: 

o) = V(mgall). 

A comparison of this with the formula w = V(g//) for the 
frequency of oscillations of a simple pendulum shows that the 
properties of the motion of a compound pendulum are the same 
as those of a simple equivalent pendulum of length 

/ = lima. 

Putting / = /„+ ma 2 (where 7 is the moment of inertia of the 
pendulum about a horizontal axis through the centre of gravity), 
we can write the equivalent length as 

/= a + Iolma. 



From this expression we can draw the following interesting 
conclusion. If we mark off OO' = / along the line OC (Fig. 33), 
and now suppose that the pendulum is suspended from an axis 



§34] DAMPED OSCILLATIONS 93 

passing through O', then the equivalent length of the resulting 
pendulum is 

r = a' + IJma'. 

But a' = l—a = hlma, and therefore /' = /. Thus the equivalent 
lengths, and therefore the periods of oscillation, are the same for 
pendulums suspended from axes at a distance / apart. 



y/////////. 



O 




Fig. 34. 

Finally, let us consider torsional oscillations of a disc sus- 
pended on an elastic wire (Fig. 34). The elastic torque which is 
created when the wire is twisted and which tends to restore the 
disc to its original position is proportional to the angle $ of 
rotation of the disc: K z = —k<\>, where k is a constant coefficient 
depending on the properties of the wire. If the moment of inertia 
of the disc about its centre is / , the frequency of oscillations is 

o>=V(*// ). 

§34. Damped oscillations 

So far we have considered the movement (including oscilla- 
tions) of bodies as if they occurred completely without hindrance. 
If a motion takes place in an external medium, however, the latter 
will resist the motion and tend to retard it. The interaction 
between a body and a medium is a complicated process which 
ultimately causes the energy of the moving body to be trans- 
formed into heat— the dissipation of energy, as it is called in 
physics. This process is not a purely mechanical one, and a 
detailed study of it involves other branches of physics also. 



94 OSCILLATIONS [iV 

From a purely mechanical point of view it can be described by 
defining a certain additional force which appears as a result of 
the motion itself and is in the opposite direction to the motion. 
This force is called friction. For sufficiently small velocities it is 
proportional to the velocity of the body: 

F fr = -bv, 

where b is a positive constant describing the interaction between 
the body and the medium, and the minus sign indicates that the 
force is in the opposite direction to the velocity. 

Let us see what is the effect of such friction on an oscillatory 
motion. We shall suppose that the friction is so small that the 
resulting energy loss by the body in one period of oscillation is 
relatively small. 

The energy loss is defined as the work done by the friction. In 
a time dt this work, and therefore the energy loss dE, is equal to 
the product of the force F fr and the displacement x = v dt of the 
body: 

dE = F tr dx = -bv 2 dt, 
whence 

dE/dt = -bv 2 = -— • hmv\ 

On the above assumption that the friction is small, we can apply 
this formula to the mean energy loss over one period, replacing 
the kinetic energy \mv 2 by its mean value also. We have seen 
in §32 that the mean value of the kinetic energy of an oscillating 
body is half its total energy E. Thus we can write 

dE/dt = -2yE, 

where y = b\lm. We see that the rate of decrease of the energy is 
proportional to the energy itself. 
Writing this relation in the form 

dE/E = d(\og e E) = -2> 'dt, 



§34] DAMPED OSCILLATIONS 95 

we find log e £ = —2yt + constant, or finally 

E = E e~ 2yt , 

where E is the value of the energy at the initial instant (t = 0). 
Thus the energy of the oscillations is reduced exponentially by 
friction. The amplitude A of the oscillations decreases with the 
energy; since the energy is proportional to the square of the 
amplitude, we have 

A = A e~ yt . 

The decrease of the amplitude is determined by the damping 
coefficient y. In a time r = 1/y the amplitude decreases by a factor 
e; this is called the time constant of the decay of the oscillations. 
Our hypothesis that the friction is slight means that r is assumed 
large in comparison with the period T = In/co of the oscillations. 
The small quantity T/t is called the logarithmic damping 
decrement. 




Fig. 35. 



Figure 35 shows a graph of the displacement as a function of 
time for the damped oscillations 



x = A cos (cot + a) 
= A Q e~ yt cos (cot + a). 

The broken lines show the decrease of the amplitude. 



96 OSCILLATIONS [IV 

Friction also affects the frequency of the oscillations. By 
retarding the motion it increases the period, i.e. decreases the 
frequency of the oscillations. When the friction is slight, however, 
this change is very small, and has therefore been ignored above; it 
may be shown that the relative change in frequency is proportional 
to the square of the small quantity y/to. When the friction is 
sufficiently great, however, the retardation may be so consider- 
able that the motion is damped without oscillation; this is called 
aperiodic damping. 

§35. Forced oscillations 

In any actual oscillating system, friction processes of some 
kind always occur. The natural oscillations which result from the 
action of an initial disturbance are therefore damped in the course 
of time. 

In order to produce undamped oscillations in a system it is 
necessary to compensate the energy losses due to friction. This 
may be done by sources of energy external to the oscillating 
system. A simple case is the action on the system of a variable 
external force F ext which varies harmonically in time: 

^ext = ^0 COS Oit, 

with some frequency w (the frequency of the natural free oscilla- 
tions of the system will now be denoted by w ). Under the action 
of this force, oscillations occur in the system, at the frequency w 
of the variation of the force; these are called forced oscillations. 
The motion of the system will then be, in general, a superposition 
of both oscillations: the natural oscillations with frequency co , 
and the forced oscillations with frequency o>. 

The natural oscillations have already been discussed; let us now 
consider the forced oscillations, and determine their amplitude. 
We write these oscillations in the form 

jc = B cos {oit — P), 

where B is the amplitude and (3 some as yet unknown phase 
difference between the external force and the oscillations which 
it causes. We have written /8 with a negative sign, i.e. as a lag in 
phase, because it will be found below that this is what in fact 
occurs. 



§35] 



FORCED OSCILLATIONS 



97 



The acceleration w of a body executing forced oscillations is 
determined by the simultaneous action of three forces: the 
restoring force —kx, the external force F ex t, and the friction 
F tT = —bv. Hence 

mw = —kx — bv + F ext . 

Dividing this equation by the mass m, using the relation k/m =&> 2 , 
and again putting b/m = 2y, we have 

w = —o) Q 2 x — 2yi> + F ex Jm. 

We shall now use a convenient graphical method of represent- 
ing the oscillations, based on the fact that x = B cos <f> (where <j> 
is the phase of the oscillation) may be geometrically regarded in 
an auxiliary vector diagram as the projection on a horizontal axis 
of a radius vector of length B at an angle <f> to this axis. [To avoid 
misunderstanding it should be stressed that these "vectors" are 
not related to the concept of a vector as a physical quantity.] 




2ywB 



v§ 



Fig. 36. 




Each term in the above equation is a periodically varying 
quantity; the frequency to is the same for each term but the phases 
are different. Let us consider, for example, the instant t = 0, when 
the phase of the external force F ext = F cos cot is zero, and so 
the quantity F ex Jm is represented by a horizontal vector of 
length FJm (Fig. 36). The quantity co 2 x = <o 2 B cos(<ot — fi) oscil- 
lates with a phase lag of /3; it is represented by a vector of length 



98 OSCILLATIONS [IV 

(o 2 B turned clockwise through an angle (3 relative to the force 
vector. The acceleration w has (as we have seen in §32) an 
amplitude co 2 B and the opposite sign to the coordinate x; it is 
therefore represented by a vector in the direction opposite to x. 
Finally, the velocity v has amplitude a>B and leads x by hir in 
phase; the quantity 2yi> is represented by a vector of length 2yo)B 
perpendicular to the vector x. 
According to the equation 

Fextl m = vv + ai 2 x + 2yu, 

the oscillation of F e Jm must equal the sum of the oscillations of 
the three terms on the right-hand side. In the graph this means 
that the sum of the horizontal projections of these three vectors 
must equal F /m. For this to be so, the vector sum of these 
vectors must evidently be equal to the vector F e Jm. The diagram 
(which gives separately the cases w > o> and w < w ) shows that 
this is true if 

{2yoiBf + B 2 (a> 2 - co 2 ) 2 = (Fjmf. 
Hence we find the required amplitude of the oscillations: 

Fjm 



B = 



V[(a> 2 -a> 2 ) 2 + 4y 2 co 2 ]' 



The same diagrams can also be used to determine the phase 
difference /3; the expression for it will not be written out here, but 
it may be noted that the angle of lag of the oscillations of x relative 
to the external force is acute or obtuse according as w < cu or 

CO > co . 

We see that the amplitude of the forced oscillations is pro- 
portional to that of the external force F , and also depends on 
the relation between the frequency co of this force and the natural 
frequency o> of the system. The amplitude of the oscillations 
reaches its maximum value when these two frequencies are 
equal (o> = co ); this is called resonance. The maximum value is 

fimax = F l2m(o y, 
and is inversely proportional to the damping coefficient y. For this 



§35] FORCED OSCILLATIONS 99 

reason the friction in the system cannot be neglected at reson- 
ance, even if it is slight. 

It is interesting to compare the value B max with the static dis- 
placement B sta that the body would have under a constant force 
F . This displacement can be obtained from the general formula 
for B by putting w = 0: B sta = F /mo> 2 . The ratio of the resonance 
displacement to this static displacement is 

BmaxlB sta = Oij2y. 

We see that the relative increase in the amplitude of the oscilla- 
tions in resonance (as compared with the static displacement) is 
given by the ratio of the frequency of natural oscillations to the 
damping coefficient. For systems with small damping this ratio 
may be very large. This explains the very great importance of the 
phenomenon of resonance in physics and technology. It is widely 
utilised in order to amplify oscillations, and always avoided if the 
resonance may cause an undesirable increase in the oscillations. 



1 


w 2 B 




CJ=OIq 




F Q /m 






2ywB 


' 


'a£B 





Fig. 37. 

The origin of the amplification of the oscillations by resonance 
may be understood by considering the relation between the 
phases of the external force F ext and the velocity v. When co ¥= <o 
there is a difference of phase, and therefore the force F ext is in the 
opposite direction to the velocity during a certain fraction of each 
period, and there is then a tendency for the motion to be retarded 
instead of accelerated. At resonance, however, the phases of 
the force and the velocity are the same (see the vector diagram, 
Fig. 37); thus the force always acts in the direction of the motion 
and continually "pushes" it. 

Near resonance (i.e. when the difference |<u — o> | is small in 
comparison with the resonance frequency w ), the formula for 



100 



OSCILLATIONS 



[IV 



the amplitude of the forced oscillations can be simplified. Putting 
in the denominator oj 2 — o> 2 = (o> + o> )(w — w ), we can approxi- 
mately replace the sum a> + co by 2cd , and also replace o> by co 
in the term 4y 2 o> 2 . The result is 



B = 



F 



2ma) V[(w — o) ) 2 + y 2 ]' 
This formula may also be written 



B 



yB, 



V[(o>-o> ) 2 + y 2 ]' 



where B max = Fj2mo) y is the maximum value of the amplitude at 



'max 

resonance 




Figure 38 shows the resonance curves of amplitude as a 
function of frequency in accordance with this formula, for various 
values of the damping coefficient y; the ordinate is the ratio 
BlB max . So long as the absolute magnitude of the difference 
o) — w is small in comparison with y, the amplitude B does not 
differ greatly from its maximum value; the amplitude begins to 
decrease considerably when |a> — o> | ~ y. For this reason the 
"width" of the resonance curve is said to be of the order of y. 
The height of the maximum (for given F ) is inversely propor- 
tional to y. Thus, the smaller the damping, the sharper is the 
resonance maximum, and the narrower is the resonance curve. 

It has been mentioned above that the motion of an oscillating 



§35] FORCED OSCILLATIONS 101 

system under the action of a periodic external force is a super- 
position of forced and natural oscillations. Neglecting the slight 
damping of the natural oscillations, we have a superposition of 
two simple harmonic oscillations, with frequencies co and oj and 
some amplitudes A and B. Near resonance, the frequencies co 
and &> are nearly equal, i.e. the difference Cl = \a> — co | is small 
in comparison with co and w . Let us find the nature of the result- 
ing motion. To do so, we use a vector diagram in which each of 
the oscillations is represented by a vector (A and B in Fig. 39). 



Fig. 39. 



As the phases of the vibrations vary with time, these vectors 
rotate uniformly with angular velocities co and co ; during one 
period T a vector makes one rotation, i.e. turns through an angle 
2n, and its angular velocity is In/T, which is just the angular 
frequency. The total oscillation is given by the geometric sum of 
the two vectors, the vector C. This vector, unlike A and B, has 
a length which varies with time, since the angle between the 
vectors A and B changes owing to the difference in the angular 
velocities <o and co . The length of C will evidently vary between 
Cmax = A + B, when the vectors A and B are in the same direction, 
and C min = \A—B\, when they are in opposite directions. This 
variation occurs periodically with frequency ft, the latter being 
the relative angular velocity of rotation of the vectors A and B. 

In the case considered, where the frequencies co and co are 
almost equal, the vectors A and B rotate rapidly while at the same 
time having only a slow relative rotation. The variation of the 
resultant vector C may be regarded as a uniform rotation with 
the same frequency co ~ co (neglecting the difference between 
co and co) together with a slow change in its length (with frequency 
ft). In other words, the resultant motion is an oscillation with a 
slowly varying amplitude. 

The periodic variation of the resultant amplitude in the super- 
position of oscillations with neighbouring frequencies is called 



102 



OSCILLATIONS 



[IV 



beats, and O, is called the beat frequency. Figure 40 shows the 
beats when A = B. 




Fig. 40. 

§36. Parametric resonance 

Undamped oscillations can be caused not only by a periodic 
external force but also by a periodic variation of the parameters 
of the oscillating system. This is called parametric resonance. As 
an example, we may consider the build-up of the oscillations of 
a swing by a person who rhythmically stands up and sits down 
and thereby periodically changes the position of the centre of 
gravity of the system. 




To elucidate the mechanism of this method of causing oscilla- 
tions, let us take the simple example of a pendulum whose length 
can be varied by pulling and releasing a string on a pulley (Fig. 
41). Let us suppose that, at each passage through the equilibrium 
(vertical) position, the pendulum is raised by the external force 
F through a short vertical distance a (small compared with the 
length / of the pendulum) and, at each extreme position, the string 



§36] PARAMETRIC RESONANCE 103 

is released the same distance a. During each period, therefore, 
the pendulum is twice lengthened and shortened, and so the 
frequency of the periodic variation of the parameter (the length of 
the pendulum) is twice that of the natural oscillations. 

Since the string is lengthened when the pendulum is deflected, 
it will descend a distance acos(f> (where </> is the angular 
amplitude of the oscillations of the pendulum), which is less 
than the height a through which it rises when the string is raised. 
In each raising and lowering the external force acting on the 
string will do an amount of work 

mga ( 1 — cos 0o ) ~ imga<l>o 2 

against gravity (since <f> is assumed small, cos <f> ~ 1 — i0 o 2 )- In 
addition, the external force F does work against the centrifugal 
force which tightens the string. In the lowest position this work 
is mv 2 ll (where v is the maximum velocity of the pendulum) 
and in the extreme positions of the pendulum it is zero, since the 
velocity is zero. Thus the total work done by the external force 
in one period of oscillation of the pendulum is 

A = 2(?mga(f) 2 + mv 2 a/l) . 

But v = /</> w, where w = V(g/l) is the frequency of the oscilla- 
tions of the pendulum; thus 

A = 6(a/l) . ?mv 2 . 

We see that the work done by the external force on the pendu- 
lum is positive and is proportional to the energy of the pendulum. 
This energy will therefore increase steadily, receiving in each 
period a small increment proportional to the energy and to the 
quantity all. 

This is the mechanism of parametric resonance. A periodic 
variation of the parameters of an oscillating system (with a period 
twice the natural frequency of the system) may bring about a 
steady increase in its mean energy E, the rate of increase being 
proportional to E: 

dEjdt = 2kE, 



104 OSCILLATIONS [IV 

where k is a (small) constant. This is a relation of the same form 
as for damped oscillations, except that the derivative dE/dt is 
positive instead of negative. This means that the energy (and 
therefore the amplitude) of the oscillations increases exponenti- 
ally with time. 

In reality, of course, there is always some friction which 
causes damping of the oscillations. Consequently, in order for 
parametric resonance build-up of oscillations to occur, the 
amplification coefficient k must exceed a certain minimum value, 
namely the damping coefficient due to friction. 

We have discussed the production of oscillations in a system by 
periodic external interaction. There are, however, oscillatory 
systems in which oscillations are caused not by a periodic force 
but by a steady source of energy, which compensates the energy 
losses in the system that bring about the damping of the oscilla- 
tions. One example of such a system is a clock, in which a 
compressed spring or raised weights act as the source of energy. 



CHAPTER V 

THE STRUCTURE OF MATTER 



§37. Atoms 

We shall not give here a detailed account of the problems of 
atomic physics, but simply describe some of the basic facts con- 
cerning the structure of matter which will be needed subsequently. 

All bodies are made up of a fairly small number of simple 
substances, the chemical elements. The smallest particle of 
each element is an atom of that element. 

The masses of the atoms are extremely small. It is therefore 
more convenient to measure them in special units, and not in 
grams. It would be natural to take as the unit the mass of the 
lightest atom, that of hydrogen. However, the precise standard 
of atomic weights is customarily taken not as the atom of hydrogen 
but as that of oxygen, which is more convenient for chemical 
purposes. The oxygen atom is approximately 16 times heavier 
than the hydrogen atom, and the unit of atomic weight is taken 
as one-sixteenth of the mass of the oxygen atom; this definition 
will be slightly refined in §38. The mass of the atom of any 
element expressed in these units is called the atomic weight 
of the element, and is usually denoted by A. The atomic weight 
of hydrogen is 1-008. 

The mass of an atom in grams is proportional to its atomic 
weight. It is therefore clear that the number of atoms in a quantity 
of any element whose mass in grams is numerically equal to 
the atomic weight of that element (called a gram-atom of the 
element) is the same for every element. This is called Avogadro's 
number, its value is 

N = 6-02 X 10 23 . 

The mass of an atom having atomic weights is clearly 

m A = A/N = 1 -66 X 10~ 24 A g. 

105 



106 THE STRUCTURE OF MATTER [V 

Although the atom is the smallest particle of an element, it 
has itself a complicated structure. An atom consists of a relatively 
heavy positively charged nucleus and a number of lighter 
negatively charged particles moving round it, the electrons, 
which form the electron shell of the atom. Different atoms have 
different nuclei, but all electrons are identical. 

The mass of an electron is so much less than that of a nucleus 
that practically all the mass of an atom is concentrated in its 
nucleus. The lightest nucleus is that of hydrogen, called a 
proton, which is approximately 2000 times (more precisely, 
1837 times) heavier than the electron. The absolute mass of 
the electron is 

m = 9-llxl0- 28 g. 

At the same time, the nucleus occupies only a negligible part of 
the volume of the atom. The radii of atoms, i.e. the radii of the 
regions round the nuclei in which the electrons move, are of the 
order of 10 _8 cm; the radii of the nuclei are tens of thousands 
of times smaller, between 10 -13 and 10 -12 cm. 
The charge on an electron is in absolute value 

e = 4.8O X 10- 10 CGSE unit of charge 
= 1-60X10- 19 C. 

It is often necessary to consider the electron charge multiplied 
by Avogadro's number, i.e. the charge on one "gram-electron". 
This product is called the fara day: 

F = eN = 9-65 X 10 4 C. 

An atom as a whole is electrically neutral, its total charge 
being zero. In other words, the positive charge on the nucleus 
is exactly compensated by the negative charge on the surrounding 
electrons. This means that the charge on the nucleus is always 
an integral multiple of the charge on an electron. We can say that 
the magnitude of the charge on the electron is an elementary 
charge; the charge on any particle existing in Nature is a multiple 
of the electron charge. This is one of the most fundamental 
physical properties of matter. 

The charge on a nucleus, expressed in units of the electron 
charge, is called the atomic number of the element and is usually 



§37] ATOMS - 107 

denoted by Z. Since the charge on the nucleus is just balanced 
by that on the electrons, the number Z is evidently also the 
number of electrons in the electron shell of the atom. All the 
properties of atoms that appear under ordinary conditions are 
determined by their electron shells, including, for example, the 
chemical and optical properties of matter. It is therefore clear 
that the atomic number is a fundamental characteristic of the 
atom, which to a considerable extent determines its properties. 
The arrangement of the elements in Mendeleev's periodic 
system is simply an arrangement in order of increasing atomic 
number, the latter being the same as the number giving the 
position of the element in the table. 

The interaction forces which determine the structure of the 
atom are mainly those of electrical interaction of the electrons 
with the nucleus and with one another: the electrons are attracted 
by the nucleus and repelled by one another. Other forces (mag- 
netic forces) are of relatively minor importance in the atom. The 
charge on the nucleus, and therefore the electric field in which 
the electrons move, are determined by the atomic number, thus 
again showing the fundamental importance of this number in 
governing the properties of atoms. 

The gravitational interaction plays no part at all in atoms. The 
energy of the electrical interaction, for example, between two 
electrons at a distance r apart is e 2 /r, and that of the gravitational 
interaction is Gm 2 /r; the ratio of these two quantities is 

Gm 2 le 2 = 2-3 X 1(T 43 . 

This number is so small that it is pointless to speak of gravitational 
interaction in the atom. 

The properties of atoms can in no way be described by means 
of classical mechanics, which is unable to explain either the 
structure of the atom or even the fact that it exists as a stable 
configuration. Classical mechanics is entirely inapplicable to 
the motions of particles of such small mass as electrons in such 
small regions of space as are occupied by atoms. Atomic phenom- 
ena can be understood only from the laws of a quite different 
mechanics called quantum mechanics. 

Under various external interactions, an atom may lose one or 
more electrons from its electron shell. We then have an atomic 



108 THE STRUCTURE OF MATTER [V 

particle which is not electrically neutral but charged, a positively 
charged ion. The energy needed to remove one outer electron 
from the atom is called the ionisation potential of the atom. 

Energies in atomic phenomena are usually measured in a 
special unit, since the erg would be too large for this purpose. 
The unit employed is the energy gained by an electron in travers- 
ing a potential difference of one volt in an electric field, and is 
called an electron-volt (eV). Since the work done by the electric 
field is equal to the product of the charge and the potential 
difference, and 1 volt is 1/300 CGSE unit of potential, we have 

1 eV = 4-80 X 10- 10 x 1/300 erg 
= 1-60X10- I2 erg. 

The ionisation potentials of atoms are measured in electron- 
volts. They range from 3-89 eV, the smallest (for caesium), to 
24-6 eV, the largest (for helium). The ionisation potential of 
the hydrogen atom is 1 3 -6 eV. 

If we consider the ionisation potential as a function of atomic 
number, we see that this function has a remarkable periodic form. 
It increases more or less steadily in each period of Mendeleev's 
table, reaching its greatest value for a noble-gas atom, and then 
drops sharply at the beginning of the next period. This is one of 
the principal manifestations of the periodic properties of atoms, 
which gave the periodic system its name. 

The ionisation potential represents the binding energy of the 
outer electrons in the atom. The inner electrons, moving deep 
in the shell, have considerably higher binding energies. The 
energy which would be needed to remove the deepest electrons 
from the electron shell reaches 10 4 or 10 5 eV in heavy atoms. 

As well as positively charged ions, negative atomic ions can 
also exist, in which an extra electron is attached to the atom. 
However, by no means all isolated atoms are capable of attaching 
an electron to form a stable system, i.e. not all atoms have an 
affinity for an extra electron. Only the halogen atoms (F, CI, Br, 
I), hydrogen and the atoms of elements of the oxygen group 
(O, S, Se, Te) can form negative ions. These elements have 
different affinities for an electron; it is greatest for the halogens 
and least for hydrogen, where the binding energy in the negative 
ion is only about 0- 1 eV. 



§38] isotopes 109 

Ions are usually denoted by the symbol of the chemical element 
with indices + or — equal in number to the charge on the ion: 
H + , Cl~, etc. Sometimes dots and primes are used: H*, CI'. 

§38. Isotopes 

The nuclei of atoms are composite structures, in general 
consisting of many particles. Their constituent parts are protons 
(hydrogen nuclei) and neutrons, which are particles whose mass 
is almost equal to that of the proton but which differ from protons 
in having no electric charge. The total number of protons and 
neutrons in the nucleus is called its mass number. Since the 
charge on the nucleus is determined by the protons present, its 
value in units of the elementary charge e is equal to the number 
of protons; that is, the number of protons in the nucleus is equal 
to the atomic number Z. The remaining particles in the nucleus 
are neutrons. 

The particles in the nucleus are held together by specific 
forces, which are not electrical. These forces are extremely 
strong, and the binding energies of the particles in the nucleus 
are tens of millions of electron-volts, and so are very large in 
comparison with the binding energies of the electrons in the 
atom. For this reason the atomic nuclei undergo no internal 
changes in any phenomena which are not of nuclear origin, and 
behave simply as particles of given mass and charge. 

We have already mentioned that the properties of an atom are 
determined mainly by the charge on the nucleus. The mass of 
the nucleus is of relatively minor importance. This fact is clearly 
evident in atoms with the same atomic number but having nuclei 
of different masses. It is found that the atoms of a given chemical 
element are not all identical; though they have the same number 
of electrons, their nuclei may differ in mass while having the 
same charge. Such varieties of a given element are called isotopes 
of the element. All the isotopes of an element are chemically 
identical, and their physical properties are also very similar. 
The number of isotopes existing in Nature is different for different 
elements, varying from one (as in Be, F, Na, Al, etc.) to ten in tin.t 



tTo avoid misunderstanding it should be emphasised that this refers only to 
the isotopes found in Nature. Other isotopes may be produced artificially, but 
their nuclei are unstable and disintegrate spontaneously. 



110 THE STRUCTURE OF MATTER [V 

The elements found on the Earth are mixtures of various iso- 
topes in definite proportions. The atomic weights given in a 
table of chemical elements are the mean weights of the atoms 
in these mixtures (often referred to as chemical atomic weights) 
and not the exact weights of any particular isotopes. The atomic 
weights of the isotopes are very close to whole numbers, the 
mass numbers, differing from them only by a few parts in a 
hundred or even a few parts in a thousand. The mean (chemical) 
atomic weights, on the other hand, of course need not be whole 
numbers. 

It is therefore necessary to refine the definition of the unit of 
atomic weight as one-sixteenth of the atomic weight of oxygen, 
given in §37. Oxygen has three isotopes: 16 0, 17 and 18 0. (The 
atomic weight, or more precisely the mass number, of an isotope 
is customarily written as an index to the left of the chemical 
symbol of the element.) The most abundant of these isotopes is 
16 0; the isotopes 17 and 18 occur in the natural mixture only 
in amounts of 0-04 and 0-2% respectively. Although these are 
relatively small quantities, they are important in a precise 
definition of atomic weights. 

The mean atomic weights of the natural mixtures of isotopes 
are usually defined relative to the atomic weight of natural 
oxygen, taken as exactly 16; this is sometimes called the chemical 
scale of atomic weights. In order to define the precise atomic 
weights of individual isotopes in nuclear physics, it is natural to 
use the atomic weight of one particular isotope as basis; the 
atomic weight of the isotope 16 is taken as 16, and the unit of 
this scale (the physical scale) is 0-027% less than that of the 
chemical scale. 

The use of two scales of atomic weights involves some in- 
convenience, and it has therefore been recommended that a 
single new scale should be employed, in which atomic weights 
are defined relative to that of the carbon isotope 12 C, taken as 
12. This change means only a very slight increase (by 0-0043%) 
in the ordinary chemical atomic weights. 

The first element in the periodic system, hydrogen, has two 
natural isotopes. As well as the principal isotope of atomic 
weight 1, there is an isotope of weight 2; natural hydrogen 
contains only one 2 H atom to about 6000 »H atoms. The "heavy" 
isotope of hydrogen is usually denoted by a separate symbol D 



§39] 



MOLECULES 



111 



and called deuterium', its nucleus is called a deuteron. Since the 
ratio of masses of the two isotopes of hydrogen is 2, which is 
comparatively large, the difference between their physical 
properties is greater than for isotopes of other elements, where 
the relative mass difference is considerably less. For example, 
"heavy water" D 2 0, containing the heavy isotope deuterium, 
freezes at 3-8°C instead of 0°C, and boils at 101 -4°C instead 
of 100°C. 

It may also be mentioned that the next element, helium, like- 
wise has two isotopes, 3 He and 4 He. The isotope 4 He is by far 
the more abundant, atoms of 3 He being present in natural helium 
only to the extent of 1 to about 10 6 atoms of 4 He. The isotope 
3 He can, however, be artificially prepared in large quantities by 
the methods of nuclear physics. 



§39. Molecules 

Atoms of different elements can combine with one another to 
form molecules. The forces of interaction between atoms which 
bring about the formation of molecules (called chemical inter- 
action), like the forces acting within the atom itself, are funda- 
mentally electrical. But the formation of molecules, like the 
structure of atoms, is a quantum phenomenon, and cannot be 
explained in terms of classical mechanics. Here we shall describe 
only the basic properties of this interaction, without investigating 
its nature. 

The simplest molecule is a diatomic molecule, consisting of two 
like or unlike atoms. The interaction of atoms which causes the 
formation of such a molecule is described by a potential energy 




Fig. 42. 



112 THE STRUCTURE OF MATTER [V 

having the form shown diagrammatically in Fig. 42. In this 
diagram the potential energy U of the interaction between the 
two atoms is plotted as a function of the distance r between them 
(more precisely, of the distance between their nuclei). This 
function has a fairly deep and sharp minimum at a certain point 
r = r . At smaller distances the curve rises very steeply; this 
region corresponds to a strong repulsion between the atoms, due 
essentially to the Coulomb repulsion between the nuclei as they 
approach. At large distances the atoms attract each other. 

The distance r corresponds to a stable equilibrium position of 
the nuclei in the molecule. In reality, the nuclei do not occupy 
exactly these positions, but oscillate about them; the amplitude 
of these oscillations, however, is usually small. The depth U 
of the potential well represents the strength of the bond between 
the atoms in the molecule; strictly speaking, the precise value of 
the binding energy needed to separate the atoms is slightly 
different from U because of the energy of the oscillations of 
the nuclei. 

The table below gives as an illustration the values of r in 
angstroms (1 A= 10 _8 cm) and U in electron-volts for some 
diatomic molecules. 



Molecule 


''o 


u 


H 2 


0-75 


4-5 


o 2 


1-2 


5-1 


Cl 2 


2-0 


2-5 


N 2 


11 


7-4 



A diatomic molecule may be likened to a dumb-bell of length 
r . Polyatomic molecules have a more complex form. Figure 43 
shows the positions of the atomic nuclei in some triatomic 
molecules, the distances being shown in angstroms. Some of 
these form triangles (water H 2 and ozone 3 ), while in others 
the atoms are collinear (carbon dioxide C0 2 and hydrocyanic 
acid HCN). In §40 examples of still more complicated molecules 
will be given. 

We see that the distances between the nuclei in molecules are 
of the order of 10 -8 cm, like the dimensions of the atoms them- 
selves. Thus the atoms in a molecule are close together, and it 



§39] MOLECULES 113 

is therefore, strictly speaking, impossible to distinguish the 
electron shells of the individual atoms in a molecule. Although 
the internal regions of the electron shells are not much changed 
when atoms combine to form a molecule, the motion of the outer 
electrons may be considerably modified, and these electrons are 
as it were shared by the atoms. 



104' 



0-96 A 



C 



H H II3A 

HjO C0 2 



H C 

• o- 




106 A 115 A 

HCN 



Fig. 43. 

In some molecules the outer parts of the electron shells are 
rearranged so that there are fewer electrons on average round 
some nuclei and more round others than in the neutral atoms; 
such molecules may be regarded as consisting of ions (for 
example, the KC1 molecule may be regarded as consisting of 
the positive ion K + and the negative ion CI - ). In other cases 
(e.g. H 2 , 2 , HC1) the atoms in the molecule remain on average 
electrically neutral. This difference, however, is only quantitative, 
and various intermediate cases between the two limiting ones 
mentioned are possible. 

A characteristic property of the chemical interaction of atoms 
is that it can be saturated. This means that the atoms which 
combine with one another because of this interaction cease to 
be able to interact in the same way with other atoms. 

Different molecules also interact; this interaction is called 
the van der Waals interaction, to distinguish it from the chemical 
interaction of atoms which leads to the formation of molecules. 

The interaction of molecules can not in general be represented 
simply by a curve U = U(r) as was done above for atoms, since 
the relative position of the molecules is described by a larger 



114 THE STRUCTURE OF MATTER [V 

number of parameters: as well as the distance r between the 
molecules, their relative orientation is important. But if the 
interaction between molecules is regarded as being averaged 
over all possible orientations, it can again be represented by such 
a curve. This curve resembles the curve of interaction between 
atoms in a molecule, in that at large distances all molecules 
attract one another, and at small distances repel one another. 
The forces of attraction between molecules decrease rapidly 
with increasing distance between them. The forces of repulsion 
between molecules increase even more rapidly as they approach, 
so that approaching molecules behave like solid bodies and do 
not interpenetrate. The depth of the minimum on the van der 
Waals interaction curve is very small, being only some tenths or 
even hundredths of an electron-volt (see §68), whereas the 
depth of the potential well on the curve of chemical interaction 
is several electron-volts. 

Another important difference between the two kinds of inter- 
action is that the van der Waals forces, unlike the chemical forces, 
do not exhibit saturation. The van der Waals interaction exists 
between all molecules, so that if two molecules are brought 
together by it they continue to attract other molecules. The 
forces of molecular attraction therefore do not lead to the forma- 
tion of "supermolecules", but merely assist the general tendency 
of all molecules to approach one another. This tendency exists 
when matter enters a condensed (liquid or solid) state. 



CHAPTER VI 

THE THEORY OF SYMMETRY 



§40. Symmetry of molecules 

The concept of symmetry plays a fundamental part in physics. 
Symmetry is one of the most important characteristics of a given 
physical object, and in many cases it has a decisive effect on the 
behaviour of the object. 

We shall begin by considering the possible symmetries of indi- 
vidual molecules. The symmetry properties consist of various 
symmetry elements, which we shall first of all define. 

A molecule is said to have an axis of symmetry of order n if it 
is left unchanged in position by a rotation through an angle lir/n 
about that axis, where n is any integer: n = 2, 3, 4, . . .; such an 
axis is usually denoted by the symbol C„. For example, if a 
molecule has an axis of symmetry of order 2, this means that the 
molecule is unchanged in position by a rotation through 180°; that 
is, each atom A, B, . . . in the molecule corresponds to another 
atom A', B',. . . of the same kind, located as shown in Fig. 44 
relative to A, B, . . . and the axis. If the molecule has an axis of 
symmetry of order 3, it is left unchanged in position by rotations 
through 120° and 240°; for each atom A, the molecule also con- 
tains two atoms of the same kind, A' and A", situated as shown 
in Fig. 45. 

A molecule has a plane of symmetry if it is left unchanged in 
position on reflection in that plane; this symmetry element is 
denoted by o\ This means that for each atom A in the molecule 
there is another atorn^' of the same kind lying on the perpendic- 
ular from A to the plane and at the same distance on the other 
side of the plane (Fig. 46). 

In addition to reflection in a plane, we can define "reflection in 
a point", giving a new symmetry element, a centre of symmetry 
or centre of inversion, denoted by i. If a molecule has a centre of 
symmetry at some point i (Fig. 47), each atom A corresponds to 

115 



116 



THE THEORY OF SYMMETRY 



[VI 



A'«- 



B >;: 



-t-^B 



Fig. 44. 



A" 






N 



Fig. 45. 



another atom A' of the same kind, lying on the line joining^ and / 
and at the same distance on the other side of i. 

Finally, there is a symmetry element called a rotary-reflection 
axis of order n, denoted by S n . A molecule possesses this sym- 
metry if it is unchanged in position by rotation through an angle 
Irr/n about an axis followed by reflection in a plane perpendicular 
to that axis. The order n of a rotary-reflection axis can only be 
even; if n is odd, say n = 3, then by repeating the rotary reflection 
6 times we easily see that the axis S n is in fact a combination of 
two independent symmetry elements, an axis of symmetry C 3 
and a plane of symmetry a perpendicular to it. If a molecule has 
a rotary-reflection axis of order 4, for example, then each atom 
A corresponds to three other atoms A', A", A'" of the same kind, 
arranged as shown in Fig. 48. The presence of such an axis 
necessarily involves the presence of a simple axis of symmetry, 
of half the order (in this case, C 2 ). 

[It may be noted that, as is easily seen, a rotary-reflection axis 
of order 2 is equivalent to a centre of symmetry at the point where 
the axis meets the plane of reflection. Thus S 2 is not a new 
symmetry element.] 




.■• A 



»A 



B' • J*-' *B 



K* 



Fig. 46. 



Fig. 47. 



§40] 



SYMMETRY OF MOLECULES 



117 



L-.---J 


^1 

1 
1 
1 


A 


9 


i i 
i i 

A' 





Fig. 48. 

These are the elements of which the symmetry of a molecule 
may be composed. The following are examples of the way in 
which various combinations of these elements occur to determine 
the symmetry of a molecule. 

The water molecule H 2 forms an isosceles triangle (Fig. 49). 
Its symmetry consists of an axis of order 2 (the altitude of the 
triangle) and two mutually perpendicular planes of symmetry 
passing through this axis C 2 . 



H* 





Fig. 49. 



Fig. 50. 



The ammonia molecule NH 3 forms an equilateral triangular 
pyramid with the N atom at the vertex and the H atoms at the 
corners of the base (the pyramid is actually very flat, its altitude 
being only about i of the edge of the base). The symmetry con- 
sists of a vertical axis of order 3 (Fig. 50) and three planes of 
symmetry passing through this axis at angles of 60°, each plane 
passing through one of the H atoms. 

A still greater number of symmetry elements occurs in the 
benzene molecule C 6 H 6 , whose atoms lie in a plane and form a 



118 



THE THEORY OF SYMMETRY 



[VI 



regular hexagon (Fig. 51). The plane containing the atoms is 
obviously a plane of symmetry, and the molecule also has an 
axis of symmetry of order 6 passing through the centre of the 
hexagon at right angles to the plane. The centre of the hexagon 
is a centre of symmetry. There are also six axes of order 2, three 
of which join diametrically opposite atoms while the other three 
bisect opposite sides of the hexagon; one axis of each kind is 
shown in Fig. 5 1 . Finally, the six planes through these axes C 2 
at right angles to the plane of the diagram are a further six planes 
of symmetry. 





Fig. 51. 



Fig. 52. 



Let us consider also the methane molecule CH 4 , which is a 
regular tetrahedron (a solid with four equal faces, each an 
equilateral triangle): the H atoms are at the four vertices, and the 
C atom at the centre (Fig. 52). This molecule has four axes of 
symmetry of order 3, each passing through one vertex and the 
centre of the tetrahedron. Three rotary-reflection axes of sym- 
metry of order 4 pass through the midpoints of opposite edges 
of the tetrahedron. Finally, there are six planes of symmetry, each 
passing through one edge and the midpoint of the opposite edge. 
Figure 52 shows one of each of these symmetry elements. 

§41. Stereoisomerism 

There is a curious effect which depends on the presence or 
absence of a certain degree of symmetry in the molecule. If a 
sufficiently asymmetric molecule undergoes a mirror reflection, 
the resulting molecule is similar to the first, but not identical 



§41] 



STEREOISOMERISM 



119 



with it. The molecule CHClBrI is of this type, for example; it is 
obtained by replacing three H atoms in the methane molecule 
CH 4 by three different atoms CI, Br and I. Figure 53 shows two 
such molecules which are mirror images in a vertical plane. It is 
obvious that these two molecules cannot be made to coincide 
by any rotation in space, and in this sense they are not identical. 
Two such similar but not identical molecules derived from each 
other by reflection are called stereoisomers. One isomer is said 
to be right-handed and the other left-handed. 





Fig. 53. 



By no means all molecules can have stereoisomers. The 
existence of these depends on the symmetry of the molecule. For 
example, if a molecule has even one plane of symmetry, its mirror 
image is identical with it, and they differ only by a rotation about 
some axis. Hence, for example, there are no stereoisomers, not 
only of the highly symmetric molecule CH 4 , but even of the much 
less symmetric molecules CH 3 C1 and CH 2 ClBr, which still 
possess a plane of symmetry. 

Similarly, molecules which have a centre of symmetry, or any 
rotary -reflection axis, have no stereoisomers. 

Stereoisomers are completely identical in almost all physical 
properties. The difference between them appears, in particular, 
in certain phenomena which occur when light passes through 
solutions of these substances (for which reason stereoisomers are 
also called optical isomers). 

The difference between stereoisomers has an important effect 
when they react with other molecules which are also asymmetric. 
The reaction between right-handed isomers of the two sub- 
stances occurs in the same way as that between their left-handed 
isomers: the two processes differ only by a mirror reflection and 
so must be identical in physical properties. Similarly, the reaction 
of a right-handed isomer with a left-handed one is the same as 



120 THE THEORY OF SYMMETRY [VI 

that of a left-handed isomer with a right-handed one. But the 
course of the reaction in the two latter cases differs considerably 
from that in the two former cases. This is the principal difference 
between stereoisomers. 

If asymmetric molecules are formed in a chemical reaction 
between two symmetric substances (not having stereoisomers), 
then, since the initial substances are unchanged by reflection, the 
same must be true of the product. The reaction therefore yields a 
mixture of equal quantities of the two isomers. 

§42. Crystal lattices 

The fundamental property of crystals is that their atoms are 
regularly arranged. It is this symmetry of the internal arrangement 
of the atoms in crystals which we shall discuss, and not that of the 
external shape of crystals. 

The set of points at which the atoms (or more precisely the 
atomic nuclei) are located is called a crystal lattice. In consider- 
ing the symmetry of the lattice we may regard it as unbounded in 
space, ignoring the boundaries of the crystal, since these do not 
affect the structure of the lattice as such. 

The fundamental characteristic of a crystal lattice is the spatial 
periodicity of its structure: the crystal as it were consists of re- 
peated units. The lattice may be divided by three families of 
parallel planes into identical parallelepipeds containing equal 
numbers of atoms arranged in the same manner. The crystal is an 
assembly of such parallelepipeds in parallel positions. Thus, if the 
lattice is moved as a whole, parallel to itself, along the direction of 
any edge of the parallelepipeds through a distance equal to an 
integral number of times the length of that edge, the lattice will be 
unchanged in appearance. Such a displacement is called a transla- 
tion, and the symmetry of the lattice with respect to these 
displacements is called translational symmetry. 

The smallest parallelepiped which can be repeated to form the 
entire crystal lattice is called the unit cell of the lattice. The size 
and shape of this cell and the arrangement of the atoms in it 
completely determine the structure of the crystal. The lengths 
and directions of the three edges of the unit cell define three 
vectors called the basic vectors of the lattice; these are the short- 
est distances through which the lattice can be displaced so as to 
remain unchanged in appearance. 



§42] CRYSTAL LATTICES 121 

If there is an atom at a vertex of a unit cell, there must evidently 
be atoms of the same kind at every vertex of every cell. The 
assembly of these like atoms in corresponding positions is called 
a Bravais lattice of the crystal in question (Fig. 54). It is a kind of 
skeleton of the crystal lattice, displaying the whole of the trans- 
lational symmetry, i.e. the complete periodicity. Any of its atoms 
can be moved to the position of any other by some translation of 
the lattice. 




Fig. 54. 

It should not be supposed that the atoms in a Bravais lattice are 
necessarily all the atoms in the crystal. They need not even be all 
the atoms of one kind. This important fact may be illustrated by 
an example, considering for clarity not a three-dimensional lattice 
(as the crystal lattice really is) but a two-dimensional one which 
is more easily shown in a diagram. 

Let the lattice consist of only one kind of atom, represented by 
the dots in Fig. 55a. It is easy to see that, although all these atoms 
are of the same kind, they are not crystallographically equivalent. 
For the fact that all the atoms in a Bravais lattice are in cor- 
responding positions means that, if any atom in it has a neighbour 
at a certain distance in a given direction, then all the atoms in the 
Bravais lattice will have neighbours of the same kind at the same 
distance in the same direction. It is therefore clear that the points 
of type 1 in Fig. 55a are not in positions of the same type as the 
points of type 2. The point 1 has a neighbour 2 at a distance d, but 
the point 2 does not have a neighbour at that distance in the same 
direction. The points 1 and 2 are therefore not equivalent and do 
not belong to the same Bravais lattice. Points of each type separ- 
ately form a Bravais lattice at a distance d from that formed by 
points of the Other type. 



122 



THE THEORY OF SYMMETRY 



[VI 



If the atoms 2 are moved to the centres of the squares formed 
by the points 1 (Fig. 55b), then all the atoms become equivalent; 
an atom 2 will have a neighbour 1 at the same distance and in the 
same direction as an atom 1 has a neighbour 2. In this structure 
all the atoms together form a single Bravais lattice. 

It is clear from the foregoing that a crystal in general consists 
of several Bravais lattices which interpenetrate one another. 
Each of these corresponds to a particular type and arrangement 
of atoms, and all the lattices, regarded simply as sets of points, 
are identical. 



• • • • 

• ••• •••• 

2 

n • • • 

&JT • • • 

«r t — -f • i» • • • 

: . p.— -> 

. i- — .* • • * — f • 

• • • 

• • • • 

• ••• •••• 



(a) 



(b) 



Fig. 55. 



If all the atoms in a crystal form a single Bravais lattice, then 
each unit cell contains only one atom. For example, in Fig. 55b 
each cell (a parallelogram in the two-dimensional lattice) contains 
a single atom 1 or 2. [Here it may be noted that in counting the 
number of atoms per cell, only one vertex of each cell is to be 
taken, the remainder being assigned to adjoining cells.] 

If, however, the crystal lattice is composed of several Bravais 
lattices, the unit cell contains more than one atom, namely one 
from each Bravais lattice. For example, in the lattice shown in 
Fig. 55a the unit cell contains two atoms: one atom 1 and one 
atom 2. 




Fig. 56. 



§43] CRYSTAL SYSTEMS 123 

The division of the crystal into basic parallelepipeds, i.e. unit 
cells, is not unique. A unit cell may, in principle, be chosen in an 
infinity of ways. To illustrate this, let us again consider a two- 
dimensional lattice (Fig. 56). We can clearly regard the unit cell 
either as the parallelogram a or as the parallelogram a' with equal 
validity. 

It is important, however, that despite this ambiguity the unit 
cell, however chosen, will contain the same number of atoms 
and have the same volume (in a two-dimensional lattice, the same 
area: the parallelograms a and a' have the same base and the 
same height, and therefore the same area). For let us consider 
atoms of a given kind and position. It is clear from the foregoing 
that each cell contains one such atom, and the number of cells in 
a volume V of the crystal is therefore always equal to the number 
N of these atoms. Thus the volume of one cell v = VI N, however 
the cell is chosen. 

§43. Crystal systems 

The Bravais lattice is a very important characteristic of a 
crystal, and the classification of the various types of crystal 
symmetry is based in the first instance on the classification of 
the various types of Bravais lattice. 

All Bravais lattices have translational symmetry. In addition 
they may also have the symmetry elements discussed in §40, i.e. 
various axes and planes of symmetry. This symmetry is the basis 
of the classification described below. 

For example, every point of a Bravais lattice is a centre of 
symmetry, since to each atom in the lattice there corresponds 
another atom collinear with that atom and the lattice point 
considered and at the same distance from this point. Thus any 
Bravais lattice has a centre of symmetry, but it may also possess 
higher symmetry. 

A body of finite size, such as a molecule, may in principle have 
an axis of symmetry of any order. A periodic structure, on the 
other hand, such as a crystal lattice, can have axes of symmetry 
only of a small number of orders: 2, 3, 4 or 6. For if the lattice 
had an axis of symmetry of order 5, say, this would mean that 
the lattice contained a plane in which there were points forming 
regular pentagons. But this is certainly impossible; the only 
regular polygons which can completely fill a plane are equilateral 



124 THE THEORY OF SYMMETRY [VI 

triangles, squares, and regular hexagons. In order to prove this 
statement, let us consider any point in the plane at which the 
sides of polygons filling the plane meet. In order to fill the plane 
completely, the angle between adjoining sides of a polygon must 
be an integral submultiple of 2tt, i.e. must be Irrlp, where p is 
any integer. The angle in a regular n-gon is 7r(n—2)ln. Thus we 
have 

TTJlt — 2) _ 2_7T 

n p ' 

whence it is seen that the quantity 2n/(n — 2) must be an integer, 
and this is true only for n — 3,4, and 6. 

Thus we see that by no means all types of symmetry are pos- 
sible in lattices. In consequence, there are only a relatively small 
number of types of symmetry of Bravais lattices. These are called 
crystal systems, and they will now be enumerated. 

1. Cubic system. The most symmetrical Bravais lattice is 
one having the symmetry of a cube. (Instead of listing the axes 
and planes of symmetry of the lattice, we shall simply state the 
geometrical figure, in this case a cube, which has the same 
symmetry.) 

This lattice is obtained by placing atoms at the vertices of 
cubic cells. There are, however, other ways of constructing a 
Bravais lattice with the symmetry of a cube. It is evident that 
the cubic symmetry is unaffected by placing an atom also at the 
centre of each cubic cell; all the atoms (at the vertices and at the 
centres of the cubic cells) will have the same relative position, 
i.e. the same neighbours, and will therefore form a single Bravais 
lattice. We can also construct a cubic Bravais lattice by adding to 
the atoms at the vertices of the cubic cells an atom at the centre 
of every face of the cubes. 

Thus there are three different Bravais lattices belonging to the 
cubic system. They are called simple, body-centred and face- 
centred lattices and denoted by the symbols P, I, F respectively. 
Figure 57 shows the arrangement of the atoms in the cells of these 
lattices. 

The cubic cell of the simple Bravais lattice is also the unit 
cell, but the cubic cells of the lattices / and F are not unit cells, 
as we see from the fact that they contain more than one atom. In 



§43] 



CRYSTAL SYSTEMS 

Cubic system 



125 



a 



* 



:^r //; 



7*. 




\// .xJtC 



I 



~X4 



Hexagonal 
system 



*■ 



Tetragonal system 



c 

a /a 



7\ &3 



Hj& Its* o 1 






/\N 



P I 

Orthorhombic system 



JX 



^ 



Rhombohedral 
system 

a. 





Monoclinic system 



\ y 



X 



Triclinic 
system 




Fig. 57. 



126 



THE THEORY OF SYMMETRY 



[VI 




Fig. 58. 

Fig. 58 the thick lines show the unit cells of all three types of 
cubic lattice. In the body-centred cubic cell there are two atoms 
(e.g. 1 and 1' in Fig. 58), and in the face-centred cell there are 
four atoms (1, 1', 1", V" in the diagram); the other atoms must be 
regarded as belonging to adjoining cells. Hence it follows that 
the volumes of the unit cells in the body-centred and face- 
centred lattices are respectively W and ia 3 , where a is the length 
of the edge of the original cube. 

The length a is called the lattice constant. It is the only numerical 
parameter that is needed to describe a cubic lattice. 

The unit cells in the body-centred and face-centred lattices 
have a form which does not itself possess the cubic symmetry 
of the lattice. In this sense the representation of the structure of 
the crystal by means of such cells does not exhibit its symmetry 
so clearly as the representation by means of the cubic cells which 
are not unit cells. The arrangement of atoms in the crystal is 
therefore usually described in terms of the cubic cells, using 
rectangular coordinates with axes X, Y, Z along three edges of 
the cubic cell and the constant a as the unit of measurement of 
the coordinates. For example, an atom at the centre of a cubic 
cell is described by the three coordinates £, i, i; the coordinates 
h i, define an atom at the centre of a face in the plane XY, and 
soon. 



■^M7\ 



Fig. 59. 



§43] CRYSTAL SYSTEMS 127 

2. Tetragonal system. If a cube is stretched in the direction 
of one of its edges, a less symmetrical geometrical figure is 
obtained, namely a right square prism, whose symmetry cor- 
responds to that of Bravais lattices of the tetragonal system. 

There are two types of such lattices, simple and body-centred, 
whose cells are shown in Fig. 57. At first sight it appears that 
we could construct a lattice with the same symmetry by adding 
to the simple lattice cell one atom at the centre of each end face 
of the prism (Fig. 59). However, it is easily seen that such a 
lattice would be reduced to another simple tetragonal Bravais 
lattice by simply taking a different basic square prism cell, so 
that no new lattice results. For by joining the atoms at the centres 
of the end faces of two adjoining cells to the atoms at the vertices, 
as shown in Fig. 59, we obtain another prism whose symmetry 
is the same as the original one and which has atoms only at the 
vertices. Similarly, there is no face-centred tetragonal Bravais 
lattice, since it is equivalent to the body-centred one. 

The tetragonal lattice is described by two constants, the edge 
length a of the base and the height c of the prism. 

3. Orthorhombic system. If a cube is stretched along two of 
its edges by different amounts, we obtain a rectangular parallele- 
piped with edges of three different lengths. The symmetry of this 
figure is that of lattices of the orthorhombic system. 

There are four types of orthorhombic Bravais lattices: simple, 
body-centred, face-centred and base-centred, the last of these 
being denoted by the symbol C. Fig. 57 shows, as for the other 
systems, the basic parallelepipeds of the orthorhombic lattices, 
whose forms correspond to the full symmetry of this system; here 
again, they coincide with the unit cell only in the simple Bravais 
lattice. 

The orthorhombic lattice is described by three parameters, the 
lengths a, b, c of the edges of the cell. These are taken as the 
units of length on the axes of rectangular coordinates along the 
corresponding edges of the cell. 

4. Monoclinic system. This has an even lower symmetry, 
namely that of the figure obtained from a rectangular parallele- 
piped by "slanting" it along one edge, giving a right parallelepiped 
with arbitrary base. This system includes two Bravais lattices, 
P and C in Fig. 57. 

The , monoclinic lattice is described by four parameters, the 



128 THE THEORY OF SYMMETRY [VI 

lengths a, b, c of the edges of the cell and the angle £ between two 
of these edges (the others being at right angles). Here again, the 
positions of the atoms are specified by means of coordinates with 
axes along the three edges of the cell, but these coordinates are 
now oblique and not rectangular. 

5. Triclinic system. This corresponds to the symmetry of an 
arbitrary oblique parallelepiped. It is the lowest symmetry, com- 
prising only a centre of symmetry. It includes only one Bravais 
lattice P, described by the lengths a,b,c of three edges of the cell 
and the angles a, /3, y between them. 

Two further crystal systems stand somewhat apart. 

6. Hexagonal system. The lattices of this system have very 
high symmetry, corresponding to that of a right regular hexagonal 
prism. The Bravais lattice of this system (denoted by H) can be 
constructed in only one way: its lattice points are at the vertices 
of hexagonal prisms and at the centres of their hexagonal faces. 

The hexagonal lattice is described by two parameters: the edge 
length a of the base and the height c of the prism. The unit cell in 
this lattice is a parallelepiped whose base is a rhombus, as shown 
by the broken lines in Fig. 57. The edges of this unit cell (height 
c and two sides a of the base at an angle of 120°) are used as 
coordinate axes in specifying the position of the atoms in the 

lattice. 

7. Rhombohedral system. This corresponds to the symmetry 
of a rhombohedron, a figure obtained by stretching or compress- 
ing a cube along one of its spatial diagonals without changing the 
length of the edges. All its faces are equal rhombuses. In the only 
Bravais lattice possible in this system (denoted by R), the lattice 
points are at the vertices of rhombohedra. This lattice is described 
by two parameters: the length a of the edges of the cell and the 
angle a between them. For a = 90° the rhombohedron becomes a 

cube. 

This completes the list of the various Bravais lattices. We see 
that there are altogether seven types of symmetry of the Bravais 
lattice, i.e. seven crystal systems, corresponding to fourteen 
different types of Bravais lattice. 

The crystal systems are the basis of the classification of crys- 
tals and are principally used to describe the properties of the 
crystal. The terms "hexagonal crystal", "cubic crystal" and so on 
frequently used for brevity must be taken as indicating the crystal 



§44] SPACE GROUPS 129 

system and not, for example, the external form of a particular 
specimen. 

It may also be mentioned that crystals of the rhombohedral, 
hexagonal and tetragonal systems, whose lattices are described 
by two parameters, are called uniaxial crystals, while those of the 
triclinic, monoclinic and orthorhombic systems are called biaxial 
crystals. 

§44. Space groups 

The Bravais lattices discussed above are sets of atoms which 
are equivalent, i.e. of the same kind and similarly situated. It has 
already been stressed that a Bravais lattice does not in general 
include all the atoms in a crystal, and an actual crystal lattice can 
be represented as an assembly of several interpenetrating Bravais 
lattices. Although all these lattices are entirely identical, the sym- 
metry of the assembly, i.e. the symmetry of the crystal itself, may 
differ considerably from that of a single Bravais lattice. 



• • • 

J • • • 

•°» «o« # ° # # o # 

• • • 

•°» . .°. «°« «° # 

£ • • 



iO, 



'• . 



• 



• • « •"• »°« . •°» 



•"• « » »°m »°» 

Fig. 60. 

This important fact may be illustrated by an example, again 
using for clarity a representation of a two-dimensional lattice. In 
Fig. 60 the white circles are the points of a two-dimensional 
"hexagonal" Bravais lattice. An axis of symmetry of order 6 
passes through each point of this lattice at right angles to the 
plane of the diagram. Now let three further lattices of the same 
kind be superposed on this lattice; their points are shown by the 
black circles in Fig. 60. It is clear that in the resulting lattice the 
axes of symmetry just mentioned will be of order 3 and not 6. 

We see that, when the actual lattice is composite, its symmetry 
may be lower than that of its Bravais lattice. 



130 



THE THEORY OF SYMMETRY 



[VI 



In actual crystal lattices it is also necessary to take into account 
the possibility of the existence of a new kind of symmetry element 
consisting of a combination of rotations or reflections with 
translations. Such elements are called screw axes and glide 
planes. 



C3 

a 

C3 



Fig. 61. 

The lattice has a screw axis of order n if it is unchanged in 
appearance by rotation through an angle 2ir\n about the axis 
together with a displacement through a certain distance along the 
axis. To illustrate this symmetry, Fig. 61 shows a linear sequence 
of atoms (to be imagined extended indefinitely in both directions) 
having a screw axis of order 3. This structure is periodic, with 
period a; it is unchanged in appearance by a rotation through 120° 
about the axis together with a displacement through \a along the 
axis. 

If the lattice is unchanged in appearance by reflection in a plane 
together with a displacement through a certain distance in a direc- 
tion lying in that plane the lattice is said to possess a glide plane. 

Thus an actual crystal has a certain translational symmetry 
(described by the type of its Bravais lattice) and may also have 



§45] CRYSTAL CLASSES 131 

simple and screw axes of symmetry, rotary-reflection axes, and 
simple and glide planes of symmetry. All these elements may be 
combined in various ways. 

The set of all symmetry elements of an actual crystal lattice is 
called its space group. This gives the most complete description 
of the symmetry of the arrangement of atoms in the crystal, i.e. 
the symmetry of its internal structure. 

There are found to be altogether 230 different space groups, 
discovered by E. S. Fedorov in 1891. These groups are custom- 
arily assigned to the crystal systems in accordance with the Bra- 
vais lattices by which they are generated. We shall not, of course, 
list here all the space groups, but merely state how they are distrib- 
uted among the systems: triclinic 2, monoclinic 13, orthorhom- 
bic 59, rhombohedral 7, tetragonal 68, hexagonal 45, cubic 36. 

The phenomenon of stereoisomerism in molecules has been 
described in §41. This can occur also in crystals (where it is 
called enantiomorphism). There exist crystals whose lattices are 
mirror images and which nevertheless cannot be made to coincide 
by any displacement in space. As with molecules, enantiomor- 
phism of crystals is possible only when the crystal lattice has no 
element of symmetry which includes reflection in a plane. An 
example of such a structure is given by crystals of ordinary 
quartz, which belongs to the rhombohedral system (this refers to 
the modification of quartz which exists at ordinary temperatures). 

§45. Crystal classes 

There are many physical phenomena in which the atomic 
structure of matter does not appear directly. In considering such 
phenomena, matter may be regarded as a continuous medium and 
its internal structure may be ignored. For example, the thermal 
expansion of solids and the deformation of solids by external 
forces are phenomena of this type. The properties of matter as a 
continuous medium are called macroscopic properties. 

The macroscopic properties of a crystal are different in different 
directions. For example, the properties of transmission of light 
through a crystal depend on the direction of the ray; the thermal 
expansion of a crystal is in general different in different directions; 
the deformation of a crystal depends on the orientation of the 
external forces, and so on. The cause of this dependence of 
properties on direction is related, of course, to the structure of the 



132 THE THEORY OF SYMMETRY [VI 

crystal. It is clear, for example, that the stretching of a cubic 
crystal in a direction parallel to the edges of the cubic cells in its 
lattice will not occur in the same way as a stretching along the 
diagonal of these cells. 

The dependence of the physical properties of a body on direc- 
tion is called anisotropy. We may say that a crystal is an aniso- 
tropic medium. In this respect crystals are fundamentally different 
from isotropic media, such as liquids and gases, whose properties 
are the same in all directions. 




Fig. 62. 

Although the properties of crystals are in general different in 
different directions, there may be some directions in which they 
are the same; such directions are said to be equivalent. For ex- 
ample, if a crystal has a centre of symmetry, any direction in it is 
equivalent to the opposite direction; if a crystal has a plane of 
symmetry, any direction is equivalent to the direction which is its 
mirror image in the plane (Fig. 62), and so on. 

It is evident that the "symmetry of directions" in a crystal, and 
therefore the symmetry of its macroscopic properties, are deter- 
mined by its axes and planes of symmetry. The translational 
symmetry is here unimportant, since a translation of the lattice 
does not affect directions in it; thus the macroscopic properties 
of a crystal do not depend on its particular Bravais lattice (among 
those possible in a given system). From this point of view it is 
also immaterial whether the crystal has a simple or a screw axis 
of symmetry of a given order, and whether a plane of symmetry is 
a simple or a glide plane. 

There is a limited number (32) of possible combinations of 
planes and axes of symmetry which can describe the symmetry of 
directions in a crystal. These combinations, i.e. types of macro- 
scopic symmetry of a crystal as an anisotropic medium, are called 
crystal classes. 



§46] LATTICES OF THE CHEMICAL ELEMENTS 133 

The relation between the space group and the class of a crystal 
is clear from the foregoing. The class is derived from the space 
group by omitting all translations and the distinctions between 
simple and screw axes of symmetry and simple and glide planes 
of symmetry. 

The crystal classes, like the space groups, are assigned to the 
systems in accordance with the Bravais lattices for which they 
actually occur in crystals. It is found that the numbers of classes 
in the systems are: triclinic 2, monoclinic 3, orthorhombic 3, 
tetragonal 7, cubic 5, rhombohedral 5 and hexagonal 7 (though it 
should be noted that all the classes in the rhombohedral system can 
be given by either a rhombohedral or a hexagonal Bravais lattice). 

Among the classes belonging to a given system there is one 
which has the full symmetry of the system. The remaining classes 
are of lower symmetry, i.e. have fewer symmetry elements than 
the system in question. 

As an example of the relation between the macroscopic pro- 
perties and the symmetry of a crystal, let us consider thermal 
expansion. 

An isotropic body (a liquid or gas) expands uniformly in all 
directions on heating, and is therefore described by a single co- 
efficient of thermal expansion. It is easily seen that the same is 
true of cubic crystals. For a crystal of the cubic system, as it 
expands, must remain a cubic crystal, and its lattice must there- 
fore retain its shape; hence it follows that such a crystal must 
expand uniformly in all directions, i.e. like an isotropic body. 

A tetragonal crystal, on the other hand, though it remains tetra- 
gonal, need not retain the same ratio of the height c and width a of 
its cells. The crystal can therefore expand differently in the direc- 
tion of the height of the cells and in directions perpendicular 
thereto. In other words, the thermal expansion of a tetragonal 
crystal is described by two coefficients (and the same is true of 
any uniaxial crystal). The thermal expansion of biaxial crystals is 
described by three coefficients which give the expansion along 
three axes. 

§46. Lattices of the chemical elements 

We shall now describe the structure of some actual crystals, 
and first mention that, although we speak for brevity of the atoms' 
being located at the lattice points, it would be more correct to say 



134 THE THEORY OF SYMMETRY [VI 

that their nuclei are located there. The atoms themselves can not 
be regarded as points in a crystal lattice; they occupy a consider- 
able volume, and neighbouring atoms are as it were in contact. 
Here, as in molecules, the outer parts of the electron shells are 
appreciably distorted and "shared", in comparison with the shells 
of isolated atoms. The most accurate and complete way of 
describing the structure of a crystal therefore consists in deter- 
mining the distribution of the "electron density" throughout the 
volume of the lattice. 

Let us first consider the crystal structure of the chemical ele- 
ments. About forty different lattices formed by the elements are 
known, and some of them are very complex. For example, one 
modification of manganese crystallises with a body-centred cubic 
Bravais lattice containing fifty-eight atoms in one cubic cell (29 
atoms in the unit cell); one modification of sulphur has a face- 
centred orthorhombic Bravais lattice with 128 atoms in one cell 
(32 atoms in the unit cell). The great majority of the elements, 
however, crystallise in comparatively simple lattices. 

About twenty elements form cubic crystals in which all the 
atoms constitute a single face-centred Bravais lattice; they in- 
clude many metals (silver, gold, copper, aluminium, etc.) and also 
the crystals of the noble gases. In the crystals of about fifteen 
elements, all of which are metals, the atoms constitute a single 
body-centred cubic Bravais lattice; these include the alkali metals 
lithium, sodium and potassium. No element, however, forms a 
simple cubic lattice. 

In order to understand the reason for this preference for body- 
centred and face-centred structures, let us consider a problem of a 
kindred type, though it has no direct physical significance: the 
packing of similar spheres. 

Let us take first the packing of spheres in a simple cubic lattice. 
Then the spheres at adjoining vertices of cubic cells are in con- 
tact, and the edge a of the cube is therefore equal to the diameter 
d of the spheres. Since each cubic cell in this lattice corresponds 
to one sphere, we can say that the volume per sphere is a 3 = d 3 . 
The volume of the sphere itself is (477-/3)</ 3 /8 = 0-52cP, i.e. is only 
52% of the volume of the cell. 

A closer packing is given by the body-centred cubic lattice. 
Here the nearest neighbours, which must be in contact, are the 
atoms at a vertex and at the centre of a cell. Since the spatial 



§46] LATTICES OF THE CHEMICAL ELEMENTS 135 

diagonal of the cube is 0V3, we must have d= aV3/2, and the 
volume of the cubic cell is therefore a 3 = 8cP/3 V3. The body- 
centred cubic cell contains two spheres, and the volume of the 
unit cell containing one sphere is 4cPj3 V3. Thus we easily find 
that the sphere occupies 68% of this volume. 

Finally, the closest packing is that given by the face-centred 
cubic lattice (for which reason it is called cubic close packing). In 
this case a sphere whose centre is at the centre of a face must 
touch spheres whose centres are at vertices of the cube. The cube 
edge length a = dv2, the volume of the unit cell is one quarter 
of the cube volume, ia 3 = J 3 /V2, and the sphere occupies 
73% of this. 




(a) 




Fig. 63. 



If this lattice is viewed along a diagonal of the cube, it can be 
seen to consist of successive layers in each of which the lattice 
points (sphere centres) form a network of equilateral triangles 
(Fig. 63a). In each successive layer the lattice points lie above the 
centres of the triangles in the previous layer, and there are three 
types of layer which alternate in succession. In Figs. 63a and 63b 
the figures indicate the correspondence between the points in 
these layers and the points of the cubic lattice. 

An equally close packing may, however, clearly be achieved by 
alternating layers of only two types (Fig. 64). This gives a hexa- 
gonal lattice with two atoms in the unit cell, called hexagonal 
close packing. In the sphere model the ratio of the height c of 
the prismatic cell in this lattice (distance between nearest 



136 



THE THEORY OF SYMMETRY 



[VI 




Fig. 64. 

similar layers) to the length a of its base edge can be shown 
by calculation to be c/a = 1-63. 

Some fifteen elements, all of which are metals, have a hexa- 
gonal close-packed lattice; they include magnesium, cadmium, 
zinc and nickel. In most of these the ratio of axes in the crystal 
is very close to the ideal value 1-63. There are exceptions, how- 
ever: in cadmium and zinc the ratio c/a is about 1-9, i.e. the lattice 
is more elongated in the direction of the prism altitude than would 
occur in close packing of spheres. This results in a more marked 
anisotropy of these crystals. 

The three types of lattice described above are those most com- 
monly found among the elements. There exist also various other 
lattices in which very few elements crystallise. Some of these will 
be described in outline below. 

The most common modification of carbon, namely graphite, 
has a hexagonal lattice; no other element crystallises in this form. 
It has a layer structure, consisting of plane parallel layers with the 
atoms at the vertices of regular hexagons (Fig. 65). The distance 




3-4 A 



Fig. 65. 



§47] 



LATTICES OF COMPOUNDS 



137 




Fig. 66. 

between adjoining layers is 2-3 times the distance between atoms 
within a layer. This explains the easy flaking of graphite. 

Another modification of carbon, diamond, has a cubic lattice 
which may be regarded as formed by two face-centred Bravais 
lattices, a quarter-diagonal of the cube apart. Thus each carbon 
atom is surrounded by four neighbours at equal distances, form- 
ing a tetrahedron. This lattice is shown in Fig. 66; the hatched 
circles and the white circles are carbon atoms forming the differ- 
ent Bravais lattices. Silicon and germanium, the homologues of 
carbon, also have lattices of the diamond type. 

The bismuth lattice is of interest. It belongs to the rhombohe- 
dral system, but is distinctive in being very nearly cubic. This 
lattice may be regarded as a slightly deformed simple cubic lattice, 
the cube being slightly flattened along its diagonal (thus becoming 
a rhombohedron), and there is also a very slight additional shift 
of the atoms. 

All the elements described above have atomic lattices, in which 
separate molecules cannot be distinguished. Some elements, 
however, crystallise in molecular lattices. For example, hydro- 
gen, nitrogen, oxygen and the halogens (fluorine, chlorine, bro- 
mine and iodine) form lattices which can be regarded as being 
composed of diatomic molecules, i.e. pairs of atoms much closer 
together than the distances between pairs. 



§47. Lattices of compounds 

The crystal lattices of chemical compounds are almost as 
various as the compounds themselves. Here we shall describe 
only some of the simplest lattices. 

One of the commonest structures is that of rock salt, NaCl: a 
cubic lattice with half the lattice points occupied by sodium atoms 



138 



THE THEORY OF SYMMETRY 



[VI 



and half by chlorine atoms (Fig. 67). Each sodium atom is sym- 
metrically surrounded by six chlorine atoms, and vice versa. The 
Bravais lattice of NaCl is a face-centred cubic lattice. Each unit 
cell contains two atoms, one of sodium and one of chlorine. 





St 


^ 




7\ 


a^j 




~^A 




<s 






9< 












A 


K 


\y 


M 


y 




\ 


\ 
\ 
\ 






\JxT 




< 


l^ 




P 6 ^ 


P* 




— 5-6 A - 





4-1 A 



@Na 
OCL 



Fig. 67. 




Fig. 68. 



The arrangement of the atoms in a crystal lattice is customarily 
described by stating their coordinates (denned as in §43). It is 
sufficient to indicate the positions of the minimum number of 
atoms from which those of the remainder can be obtained by add- 
ing a lattice vector. For example, the structure of NaCl is des- 
cribed by the coordinates of two atoms relative to the axes of the 
cubic cell: Na(0,0,0), Cl(iii). The coordinates of all other 
atoms are obtained from these by adding (or subtracting) a cer- 
tain number of basic lattice vectors, which may be taken, for 
example, as the distances from the origin to the centres of the 
three faces of the cube (the points with coordinates (0,iA), 
(i (U), (U, 0)). 

The lattice of caesium chloride, CsCl, is also of a very common 
type (Fig. 68). It has a simple cubic Bravais lattice. Atoms of one 
kind are at the vertices of the cubic cells, and atoms of the other 
kind are at the centres of the cells. 

We may also mention the lattice of zinc blende, ZnS. This is 
obtained from the diamond lattice described in §46 by placing 
different atoms (Zn and S) at the points of the two interpenetrat- 
ing face-centred Bravais lattices (the hatched and white circles in 
Fig. 66). Each zinc atom is surrounded by four sulphur atoms at 
the vertices of a tetrahedron, and vice versa. The positions of the 
atoms in the cubic cell are given by the coordinates Zn(0, 0, 0), 

S\4, 4, 4J- 



§48] CRYSTAL PLANES 139 

A characteristic property of these lattices is that molecules of 
the compounds cannot be distinguished as particular groups of 
atoms. The whole crystal is, as it were, one huge molecule. 

The distribution of electrons in these lattices is such that 
around some nuclei there are on average more electrons, and 
around others fewer, than would be present in the free neutral 
atom. Such lattices may be quite adequately described as consist- 
ing of ions, and are therefore called ionic lattices. For example, 
the NaCl lattice consists of positive ions Na + and negative 
ions CI - . 

There are other lattices of compounds in which individual 
molecules can be distinguished as especially closely arranged 
groups of atoms; these include, in particular, many organic crys- 
tals. But the division of crystals into atomic and molecular is 
largely arbitrary, and various intermediate cases are possible. 

A typical example of this is the Cdl 2 lattice, which has a kind of 
layered structure. On either side of each layer of cadmium atoms 
and close to it, there is a layer of iodine atoms, the distance be- 
tween such "triple" layers being greater. Although the latter sug- 
gests a molecular composition of the substance, it is not possible 
to distinguish individual molecules within each layer. 

§48. Crystal planes 

In the study of crystals it is frequently necessary to consider 
various planes passing through the lattice. These may be planes 
forming natural faces of the crystal, or planes having certain 
physical properties: for example, if a crystal is cleaved with a 
knife, the cleavage usually occurs along particular planes having 
distinctive properties. Finally, a consideration of various planes 
in the lattice is necessary in structural analysis by the use of 
X-rays. 

It is clear that only planes which pass through atoms in the 
crystal (i.e. through its lattice points) can have particular physical 
properties. Such planes are called crystal planes, and it is these 
which we shall now discuss. 

It has already been mentioned in §43 that, in describing crys- 
tals, use is made of a coordinate system (in general oblique) 
whose axes are related in a definite way to the edges of the 
Bravais lattice cell, the coordinates being measured in terms of the 
lengths a, b, c of these edges (which are in general different). Let 



140 



THE THEORY OF SYMMETRY 



[VI 



these coordinates be denoted by x, y, z. The coordinates of the 
Bravais lattice points are given by integers (or half -integers, but 
this does not affect the subsequent discussion). 
The general equation of a plane is 

Ix + my + nz = k 

in either rectangular or oblique coordinates. If /, m, n and k are 
integers, this equation, regarded as a single equation for the three 
unknowns x, y, z, has an infinity of integral solutions. In other 
words, the plane Contains an infinity of lattice points, and is 
therefore a crystal plane. 

The significance of the numbers /, m, n is easily seen. Putting 
y = z = in the equation, we find x = kjl; this is the coordinate of 
the point where the plane intersects the x axis. Similarly, we find 
that the intercepts of the plane on the y and z axes are kjm and 
kin. Hence we conclude that the lengths of the intercepts on the 
three axes are in the ratios 

l//:l/m:l/n, 

i.e. are inversely proportional to the numbers /, m, n. These 
lengths are measured in terms of a, b, c; in ordinary units the 
lengths are in the ratios 

a\l:b\m\c\n. 

Thus we see that the numbers l,m,n determine the direction of 
the plane, i.e. its orientation relative to the axes of the lattice; 
the number k depends not on the direction of the plane but on its 




Fig. 69. 



§48] 



CRYSTAL PLANES 



141 




Fig. 70. 

distance from the origin. By giving k various integral values, with 
fixed values of /, m, n, we obtain a family of parallel crystal planes. 
It is the direction of a crystal plane which is of importance, and 
not its absolute position in the lattice. In this sense the plane is 
fully defined by the set of three numbers /, m, n. The highest com- 
mon factor may also be cancelled from these numbers, since the 
direction of the plane is obviously unchanged by this. The num- 
bers l,m,n thus defined are called the indices of the crystal plane 
and are written in parentheses: (Imn). 

As examples, we shall consider various planes in a cubic lattice. 
The plane perpendicular to the x axis (Fig. 69) has intercepts 
1, oo, oo on the axes; the reciprocals of these are 1, 0, 0, and the 
indices of the plane are (100). Similarly, the indices of the planes 
perpendicular to the y and z axes are (010) and (001). These 
planes bound a body of cubical shape and are therefore often 
called cube planes. 




Fig. 71. 



142 THE THEORY OF SYMMETRY [VI 

A diagonal plane parallel to the z axis has equal intercepts on 
the x and y axes (Fig. 70a). Its indices are therefore (110). Such 
diagonal planes are called rhombic dodecahedron planes, from 
the name of the dodecahedron bounded by planes of this kind 
(Fig. 70b). 

A diagonal plane of the cube (Fig. 71a) has equal intercepts on 
all three axes, and its indices are therefore (111). Planes of this 
kind are called octahedron planes, from the regular octahedron 
with triangular faces which they form; the octahedron shown in 
Fig. 71b is obtained by joining the centres of the six faces of the 
cube. 

§49. The natural boundary of a crystal 

The planes which form the boundaries of a natural crystal 
always pass through atoms in its lattice, and are therefore crystal 
planes. The directions of the various faces of the crystal and the 
angles between them are related to the structure of its lattice and 
are therefore characteristic of any given substance. 

Let us consider any two faces of the crystal, with indices (Imn) 
and (/' m ' n ' ). We denote by A, B , C and A ' , B ' , C ' the intercepts of 
these planes on the coordinate axes. According to the discussion 
in §48, the ratios of these intercepts (measured in ordinary units) 
are 

A:B:C = all:blm:c/n, A' :B' :C = all' :bjm' :c/n'. 

Dividing these ratios, we obtain 

AlA':BlB':ClC' = l'll:m'lm:n'ln. 

On multiplying by the least common multiple of /, m, n, the right- 
hand side of this relation is converted to the ratios of three 
integers. 

Thus we see that the ratios of the intercepts of any face of the 
crystal on the axes, when expressed in terms of the intercepts 
of any other face, are always ratios of integers. This is called the 
law of rational indices. 

The surfaces of ionic crystals must necessarily contain ions of 
different signs. Crystal planes containing ions of one sign only can 
not be crystal faces. This often provides an explanation of certain 
properties of the crystallisation of various substances. 



§49] THE NATURAL BOUNDARY OF A CRYSTAL 143 

Let us consider, for example, the NaCl crystal, whose lattice 
is shown in Fig. 67 (§47). The diagram shows how the Na + and 
Cl~ ions are situated in the (100) and (111) planes of this lattice. 
We see that the (111) plane (the diagonal plane shown by the 
broken lines in Fig. 67) passes through ions of one kind only, and 
this plane therefore cannot be a crystal face. Thus rock salt can- 
not crystallise in octahedra. The (001) plane (the cube face in 
Fig. 67), however, contains ions of opposite sign alternating in 
both directions; thus rock salt can crystallise in cubes. 

In the caesium-chloride lattice (Fig. 68, §47), on the other 
hand, the (100) planes contain ions of only one sign, and this 
substance therefore can not crystallise in cubes. 

The nature of the external boundary of a crystal, like all its 
macroscopic properties, depends on the crystal class. Thus a 
study of the shape of natural crystals enables us, in principle, to 
determine their symmetry classes. In practice this may be ren- 
dered difficult by irregularities of shape due to various accidental 
effects of the conditions in which the crystal was grown. Further 
information may be obtainable from the artificial formation of 
new faces by etching the surface of the crystal with a solvent. 



CHAPTER VII 

HEAT 

§50. Temperature 

In all bodies existing in Nature there is a continual movement 
of their constituent particles. This movement is universal: the 
molecules are moving, and so are the atoms within them. The 
characteristic feature of this movement is the randomness which 
it always to some extent possesses. The movement is called 
thermal motion, and it is the underlying cause of the phenomena 
of heat. 

Although usually the term "thermal motion" refers to the mo- 
tion which takes place on an atomic or microscopic scale, thermal 
motion is also a property of larger, macroscopic, particles. A 
well-known example of this is the Brownian motion, the random 
motion of fine particles suspended in a liquid, which may be 
observed through a microscope. 

If two bodies are brought into contact, the atoms in them will 
collide and transfer energy to one another. Thus, when two 
bodies are in contact, energy passes from one body to the other; 
the body which loses energy in this process is said to be the 
hotter, and that which gains energy the colder body. This trans- 
fer of energy continues until a definite state of thermal equilibrium 
is set up. 

To describe the hotness of bodies, the concept of tempera- 
ture is used. A quantitative definition of this might in principle 
be given by using any property of bodies which depends on their 
hotness. For example, we could define a scale of temperature 
simply by the volume of a column of mercury in thermal equi- 
librium with the body concerned. It is evident, however, that such 
a temperature scale would be entirely arbitrary and would have 
no particular physical significance; the temperature thus defined 
would be extremely inconvenient for the quantitative description 
of any other thermal phenomena. It is therefore necessary to 
establish first of all a temperature scale having a physical signifi- 

144 



§50] TEMPERATURE 145 

cance and not dependent on the nature of any one material, 
such as mercury or the glass of the vessel which contains the 
mercury. 

In physics the thermodynamic or absolute scale of temperature 
is used; it is intimately related to the general thermal properties of 
all bodies. It cannot be precisely defined here, since this would 
require a theoretical analysis of thermal phenomena which is out- 
side the scope of this book. Instead, we shall describe the scale by 
means of some of its "secondary" properties. 

It is clear that a physical definition of temperature must be 
based on a physical quantity which describes the state of a body 
and which is necessarily the same for any two bodies in thermal 
equilibrium. The mean kinetic energy of the translational motion 
of the particles (molecules or atoms) in a body is in fact found to 
have this remarkable property. If the mean energies are the same 
for the particles in any two bodies, then, when the bodies are 
brought into contact, individual particles will transfer energy in 
both directions, but there will be no net transfer of energy from 
either body to the other. 

For this reason, the mean kinetic energy of the translational 
motion of the particles within the body may be taken as a measure 
of temperature. The temperature T is customarily defined as two- 
thirds of this energy: 

T = \-\mv 1 — %mv 2 . 

Here m is the mass and v the velocity of a particle, and the bar 
over an expression denotes that its mean value is to be taken. 
(The mean value may be understood as the mean energy of vari- 
ous particles in the body at a given instant, or as the mean energy 
of a given particle at various instants, the two definitions being 
entirely equivalent.) 

According to the above definition, temperature has the dimen- 
sions of energy, and may therefore be measured in the same units 
as energy, such as ergs. The erg, however, is an extremely in- 
convenient unit for the measurement of temperature, mainly be- 
cause the energy of the thermal motion of particles is usually very 
small in comparison with the erg. Moreover, the direct measure- 
ment of temperature as the energy of particles would of course be 
very difficult to carry out in practice. 



146 HEAT [VII 

For these reasons, a conventional but convenient unit of tem- 
perature, the degree, is used in physics. It is denned as one- 
hundredth of the difference between the boiling point and the 
freezing point of pure water at atmospheric pressure. 

The conversion factor which determines the degree as a frac- 
tion of the erg is called Boltzmann's constant, and is usually 
denoted by k. Its value is 

k= l-38xl0- 16 erg/deg. 

We see that the degree is in fact very small compared with the 
erg. As a further illustration we may find the change in the total 
kinetic energy of the particles in one gram-molecule of matter 
which corresponds to each degree of temperature change. This is 
obtained by multiplying k by Avogadro's number N : 

kN = 1 -38 x 10- 16 x 6-02 x 10 23 erg 
= 8-31 J. 

We may also give the conversion factor between the degree and 
the electron-volt, the latter being the unit of energy generally used 
in atomic physics: 

leV= l-60xl(T 12 erg 

= 1-60X10-' 2 n6 

1-38 X10- 16 & B 

In what follows we shall always denote by T the temperature 
measured in degrees. The temperature measured in ergs is then 
kT, and so its definition given above must be written 

kT = %mv 2 . 

Since the kinetic energy is positive, so is the temperature T. 
It should be emphasised that this property of the temperature is 
not to be regarded as a law of Nature; it is simply a consequence 
of the definition of temperature. 

As already mentioned, the scale of temperature thus defined is 
called the absolute scale. The zero of temperature on this scale is 



§50] TEMPERATURE 147 

the temperature at which the thermal motion ceases entirely. The 
scale of absolute temperature measured from this absolute zero is 
called the Kelvin scale, and degrees on it are denoted by the 
symbol °K. 

Besides the Kelvin scale, another scale is widely used in prac- 
tice, in which the temperature is measured from the freezing 
point of water, arbitrarily taken as the zero of temperature. This 
is called the Celsius scale, and degrees on it are denoted by the 
symbol °C. 

To convert temperatures from one scale to the other, it is 
necessary to know the absolute temperature of the freezing point 
of water. According to recent measurements this is 273-15°K. 
Correspondingly, on the Celsius scale absolute zero is — 273-15°C. 

In the following, T will always denote the absolute tempera- 
ture; the Celsius temperature, if needed, will be denoted by t. 
Clearly T=t + 273-15°. 

An experiment is often said to be conducted at room tempera- 
ture, meaning 20°C (i.e. about 293°K). It is useful to note that 
this temperature measured in electron-volts is about 1/40 eV. 

To describe the velocity of the thermal motion of particles, we 
can use the square root of the quantity W which appears in the 
definition of temperature; this square root is usually called 
simply the thermal velocity and denoted by v T : 

v T =Vv*=V(3kT/m). 

This formula determines the thermal velocity of an atom, a mole- 
cule or a Brownian particle, according to the mass that is sub- 
stituted in it. When the application is to molecules, it is convenient 
to modify the formula somewhat by multiplying the numerator 
and denominator of the radicand by Avogadro's number and 
using the fact that the product mN is the molecular weight /x of 
the substance: 

v T = V(3N kTlfjL) 

= 15-8 X 10 3 V(77/a) cm/sec. 

For example, the thermal velocity of molecules of hydrogen 
(H 2 , fx = 2) at room temperature, is 1-9X 10 5 cm/sec, i.e. about 
2 km/sec. 



148 HEAT [VII 

We see that the thermal velocity is proportional to the square 
root of the temperature and inversely proportional to the square 
root of the mass of the particles. The latter relation is the reason 
why the thermal motion, which is very violent for molecules, is 
still appreciable for the microscopically small particles in the 
Brownian motion, but entirely negligible for massive bodies. 

Let us return to the definition of temperature given above. It 
must be emphasised that this definition is based on classical 
mechanics. The quantitative relation which it asserts between the 
temperature and the energy of the thermal motion of the particles 
is valid only so long as this motion can be described by classical 
mechanics. It is found that, as the temperature decreases and the 
particle energy diminishes, the conditions for classical mechanics 
to be valid are eventually no longer satisfied, and classical 
mechanics must be replaced by quantum mechanics. This occurs 
sooner for particles of smaller mass and for those whose motion 
is more restricted by the forces acting. For example, the mole- 
cules of gas in translational motion move almost as free particles, 
and this motion can always be treated classically, but the motion 
of atoms in the molecule is of the nature of small oscillations in a 
"potential well" around certain equilibrium positions, and classi- 
cal mechanics very soon ceases to be applicable to this motion. 
We shall return to this subject in §§57 and 58. 

It has been mentioned above that the thermal motion no 
longer occurs at absolute zero. This does not mean, however, 
that all motion of the particles in a body has stopped. According 
to quantum mechanics, the motion of the particles never ceases 
completely. Even at absolute zero there must remain some vibra- 
tional motion of the atoms within molecules, or vibrations of the 
atoms about the crystal lattice points in a solid. This motion, 
called zero-point vibrations, is a quantum phenomenon, and its 
energy is a measure of the quantum nature of a given object. A 
comparison of the energy of the thermal motion of the particles 
with the energy of their zero-point motion may serve as a criterion 
of the applicability of classical mechanics: the latter is suitable 
for the description of the thermal motion of the particles if the 
energy of this motion is sufficiently large compared with the zero- 
point energy. 

The most striking instance of the zero-point motion, which is 
fully maintained even at absolute zero, is the motion of electrons, 



§51] PRESSURE 149 

the lightest particles, in atoms. The motion of electrons within 
the atom is always a purely quantum phenomenon. Owing to its 
relatively high energy, it is affected only to a very slight extent by 
the temperature of the body. The thermal motion of the atoms has 
a considerable effect on their electron shells only at very high 
temperatures, of the order of many thousands of degrees. 

§51. Pressure 

On account of the thermal motion of its particles, a gas (or 
liquid) exerts a pressure on the walls of the vessel containing it. 
The gas molecules, on colliding with the walls, transfer some 
momentum to them, and the change in momentum of a body per 
unit time defines the force acting on it. 

The force exerted by the gas (or liquid) per unit area of the 
wall gives the pressure on the wall of the vessel, which will be 
denoted by p. The dimensions of pressure are those of force 
divided by those of area, and can be written in various ways: 



cm' cm" 5 cm.sec^ 

In particular, it should be noted that the dimensions of pressure 
are the same as those of energy per unit volume. 

The unit of pressure in the CGS system is 1 dyn/cm 2 : a force 
of one dyne acting on an area of one square centimetre. This unit 
is very small, however. A unit 10 6 times larger is called a bar: 

1 bar = 10 6 dyn/cm 2 = 10 N/m 2 . 

The pressure at which a force of 1 kgf acts on an area of 1 cm 2 
is called a metric or technical atmosphere (at): 

1 at = 1 kgf/cm 2 = 0-981 bar. 

The standard atmosphere (atm) is the pressure of a column of 
mercury of height 760 mm (with a certain density of mercury and 
a standard acceleration due to gravity): 

latm= 1-013 bar = 1-033 at. 



150 HEAT [VII 

The pressure corresponding to one millimetre of mercury is 
lmmHg= 1-333 X lO" 3 bar. 

The properties of bodies taken as a whole without considering 
the details of their molecular structure (on which these properties 
in fact depend) are called macroscopic properties. Temperature 
and pressure are among the most important quantities describing 
the macroscopic state of a body; another such quantity is the 
volume of the body (denoted by V). These three quantities, 
however, are not independent. For example, if a certain quantity 
of gas is enclosed in a vessel of given volume and has a given 
temperature, its pressure is thereby determined; if the volume 
or the temperature is changed, the pressure of the gas also changes. 

Thus only two of the three quantities p,V,T can be arbitrarily 
specified, the third being a function of these. We may say that 
the thermal properties are entirely determined by specifying any 
two of these quantities. 

The functional relation between the pressure, volume and 
temperature of a body is called the equation of state of the body 
concerned, and is one of the most important relations describing 
its thermal properties. 

The theoretical form of this relation can be established only 
for the simplest substances (see §53). In practice, therefore, 
experimental measurements must be used, the results of which 
can be represented graphically. Since a relation between three 
quantities is concerned, it would be fully represented by a surface 
in a three-dimensional coordinate system with p, V and T plotted 
along the axes. However, since a three-dimensional construction 
is inconvenient in practice, only two-dimensional diagrams are 
generally drawn, showing families of curves which are the inter- 
sections of the surface with various planes parallel to one of the 
coordinate planes. For example, by taking the intersections of 
the surface with planes parallel to the pV plane, i.e. perpendicular 
to the T axis, we obtain a family of curves called isotherms, 
which give the pressure as a function of the volume of the body 
for various given values of the temperature. Similarly isobars 
can be drawn; these are curves which give V as a function of T 
for given values of p. Finally, isochores give p as a function of 
T for given values of V. 



§52] AGGREGATE STATES OF MATTER 151 

It has already been mentioned in §50 that the exchange of 
energy between bodies in contact continues until their tempera- 
tures are equal and thermal equilibrium is reached. A state of 
thermal equilibrium of a system of bodies is defined as a state in 
which no spontaneous thermal processes occur in the system and 
every part of the system is at rest relative to the other parts and 
has no macroscopic motion (as opposed to the microscopic ther- 
mal motion of the particles within bodies). We may now add that in 
equilibrium not only the temperatures but also the pressures of all 
bodies in contact must be equal, since otherwise the total forces on 
the bodies would not be zero and the bodies would begin to move. 

Under ordinary conditions the pressure is positive, i.e. is in the 
same direction as if the body were tending to expand. This is not 
necessary, however, and states of negative pressure are also 
possible, in which the body is as if it were "stretched" and there- 
fore tends to contract. "Stretched" states of a liquid can be 
brought about by sealing a carefully purified heated liquid in a 
thick- walled capillary. When the capillary cools, if its walls 
contract more slowly than the liquid, the latter should occupy 
only part of the volume within the capillary. The liquid, however, 
adheres to the walls and is thus "stretched" over the whole 
volume of the capillary. Another method is to place a liquid in a 
glass capillary open at each end, which is then rapidly rotated 
about its midpoint. The liquid is stretched by centrifugal forces 
and, when a certain speed of rotation is reached, it finally "breaks" 
and is thrown out of the capillary. Considerable negative pres- 
sures can be attained by these methods: up to 280 atm in water 
(at room temperature), up to 40 atm in alcohol, up to 160 atm in 
benzene, and so on. These values may be regarded as representing 
the resistance of the liquid to disruption. 

§52. Aggregate states of matter 

The concept of aggregate states (gaseous, liquid and solid) is 
used to give the most general description of the thermal properties 
of bodies. 

Owing to the low density of matter in the gaseous state, its 
molecules are relatively far apart, being at distances large com- 
pared with the size of the molecules themselves. The interaction 
between the molecules of a gas is therefore of subordinate im- 
portance, and for the greater part of the time the molecules move 



152 HEAT [VII 

freely, undergoing collisions with one another only quite rarely. 
In liquids, on the other hand, the molecules are at distances 
comparable with their own dimensions, so that they are all in 
continual strong interaction and their thermal motion is highly 
complicated and irregular. 

Under ordinary conditions, liquids and gases differ so greatly 
in density that there is no difficulty in distinguishing between 
them. Nevertheless, the difference between these two states of 
matter is in fact not fundamental, but merely quantitative, arising 
from the value of the density and the consequent degree of 
interaction of the molecules. The lack of any fundamental distinc- 
tion between them is especially clear from the fact that the transi- 
tion between a liquid state and a gaseous state can, in principle, 
be completely continuous, so that there is no instant at which we 
can say that one state ceases to exist and the other commences. 
This will be discussed further in §69. 

The difference between liquids and what are called amorphous 
(non-crystalline) solids is also quantitative; the latter substances 
include glass, various resins, etc. Here again the absence of any 
fundamental difference is shown by the possibility of a continuous 
transition from one state to the other, achieved simply by heating. 
For example, solid glass, when heated, becomes gradually softer 
and finally entirely liquid; this process is completely continuous 
and there is no "instant of transition". The density of an amor- 
phous solid is not greatly different from that of the liquid formed 
from it. The main quantitative difference between them is in 
viscosity, i.e. the ease with which they flow; this will be further 
discussed in §118. 

A general property of gases, liquids and amorphous solids is 
that the molecules in them are randomly distributed. This brings 
about the isotropy of these bodies, i.e. the fact that their proper- 
ties are the same in all directions. The property of isotropy makes 
these bodies fundamentally different from the anisotropic crystal- 
line solids, in which the atoms are arranged in a regular manner. 

The thermal motion of the atoms in solids consists of small 
oscillations about certain equilibrium positions. In crystals, these 
positions are the crystal lattice points; in this respect the discus- 
sion in Chapter VI was imprecise in that it referred to the lattice 
points as the positions of the atomic nuclei, instead of the points 
about which the nuclei oscillate. Although the thermal motion in 



§53] IDEAL GASES 153 

solids is more "ordered" than in gases or liquids (the atoms 
remaining close to the lattice points), it is random in the sense 
that the amplitudes and phases of the various atoms are entirely 
unrelated. 

Almost all solid bodies are crystalline, but only rarely are they 
separate crystals regular throughout their volume, called single 
crystals; such crystals are formed only under special conditions 
of growth. 

Crystalline solids usually exist as polycrystals; all metals, for 
example, are of this kind. Such bodies consist of a very large 
number of individual crystallites or grains, often of microscopic 
size; for example, the dimensions of the crystallites in metals 
are usually of the order of 10 -5 to 10~ 3 cm (the size depending 
considerably on the methods by which the metal is produced 
and treated). 

The relative position and orientation of the individual crystal- 
lites in a polycrystalline substance are usually entirely random. 
For this reason, when such a substance is considered on a scale 
large in comparison with the dimensions of the crystallites, it is 
isotropic. It is clear from the foregoing that this isotropy of 
polycrystalline bodies is only secondary, in contrast to their true 
molecular anisotropy which takes effect in the anisotropy of 
individual crystallites. 

As a result of some particular treatment or a special method 
of growth, it may be possible to prepare a polycrystalline sub- 
stance in which the crystallites have a preferred crystallographic 
orientation. Such substances are said to have texture. For example, 
texture may be produced in metals by various kinds of cold 
working. The properties of these substances are, of course, 
anisotropic. 

§53. Ideal gases 

The simplest thermal properties are those of a gas so rarefied 
that the interaction between its molecules is of no practical 
importance. Such a gas, in which the interactions between the 
molecules may be neglected, is called an ideal gas. 

It should not be thought that the interaction between the 
molecules of an ideal gas does not exist at all. On the contrary, 
its molecules collide with one another and these collisions are 
important in bringing about the particular thermal properties of 



154 HEAT [VII 

the gas. But the collisions occur so rarely that the gas molecules 
move as free particles for the greater part of the time. 

Let us derive the equation of state of an ideal gas, i.e. the 
relation between its pressure, volume and temperature. To do so, 
we imagine the gas to be enclosed in a vessel having the shape 
of a rectangular parallelepiped, and assume that the walls are 
"perfectly reflecting", i.e. reflect the incident molecules at an 
angle equal to the angle of incidence, without change in the 
magnitude of the velocity. (In Fig. 72 v and v' are the velocities 
of a molecule before and after the collision; they are equal in 
magnitude and are at the same angle a to the normal to the wall.) 
These assumptions are made for simplicity; it is evident that the 
internal properties of the gas as such cannot in fact depend either 
on the shape of the vessel or on the properties of its walls. 




mmmmmmm* 

Fig. 72. 



The pressure of the gas on one face of the parallelepiped may 
be found by determining the momentum transferred to this face 
per unit time by molecules colliding with it. Since only the 
velocity component v z perpendicular to the surface of the wall 
changes in a collision, and the change is simply a change in sign, 
the momentum transferred in one collision is mv z —(—mv z ) = 2mv z , 
where m is the mass of a molecule. When moving freely, the 
molecule traverses the distance (h, say) between opposite walls 
in a time h/v z , and so it returns to the first wall in a time 2h\v z . 
Thus each molecule has vjlh collisions with a given wall per 
unit time and transfers to it a momentum 2mv z . vjlh = mv z 2 /h. 
The total force F z acting on the wall is the momentum transferred 
to it per unit time by all the gas molecules, 



F *=i2 



mt). 



where 2 denotes summation over all the molecules. 



§53] IDEAL GASES 155 

If the number of molecules in th e ves sel is N, the sum can be 
written as N times the mean value mv z 2 . But since all directions 
are c o mplete ly equivalent with respect to the gas itself, we have 
mv x 2 = mvy = mv 2 and, since v x 2 + v y 2 + v 2 = v 2 , 



Thus we have 



mv z 2 = %mv 2 . 



17 ! W — 
r, = -, — r-rair. 



h 3 

Replacing F z by pS, where p is the pressure of the gas and S the 
area of the face, and noting that hS is the volume V of the 
parallelepiped, we have 

pV = %Nmv 2 = f A/ . \mv 2 . 

The mean kinetic energy of a molecule is, from the definition of 
temperature, \kT, and thus we have finally the equation of state 
for an ideal gas: 

pV=NkT. 

This is a universal equation, involving no quantities dependent 
on the nature of the gas — a result which is an obvious effect of 
neglecting the interaction between the molecules and thus 
depriving the gas of any "individuality". 

For two different ideal gases occupying equal volumes at the 
same pressure and temperature, the number of molecules will be 
the same in each gas. This is Avogadro's law. In particular, one 
cubic centimetre of any ideal gas under standard conditions, i.e. 
at temperature 0°C and pressure 1 atm, contains 

1*013 x 10 6 x 1 
L = pV/kT = 1.38x10-^x273 = 2 ' 7 x 1Q19 mol eciiles; 

this is sometimes called Loschmidt's number. 

The number N of molecules in a given mass of gas may be 
written JV = vN , where v is the number of gram-molecules 



156 HEAT [VII 

(moles) of gas and N is Avogadro's number. Then the equation 
of state becomes 

P V=vRT, 

where R = kN is called the gas constant. In particular, for one 
mole of gas we have 

pV = RT. 

Multiplication of the values of k and N gives 

r = 8-314 X 10 7 erg/deg.mole 
= 8-314J/deg.mole; 

if the calorie is used as the unit of energy, R is very nearly equal 
to 2 cal/deg.mole. 

If the gas pressure is measured in atmospheres and the pressure 
in litres, then 

R = 0-082 l.atm/deg.mole. 

Using this value, we can easily find the volume of one gram- 
molecule of gas at 1 atm pressure and 0°C: 

V = RTlp = 0-082 X 273/1 = 22-4 1. 

At constant temperature the product of the pressure and volume 
of a given quantity of gas is constant: 

pV= constant for T = constant. 

This is Boyle's law. 

From the equation of state of an ideal gas it also follows that, 
if a certain mass of gas is at constant pressure, its volume is 
proportional to the absolute temperature of the gas: 

V/Vo = T/T for p = constant, 

where V and V are the values of the gas volume at temperatures 
T and T - Similarly 

plp =T/T for V= constant. 



§54] AN IDEAL GAS IN AN EXTERNAL FIELD 157 

These important relations show that the absolute scale of tem- 
perature can be constructed without measuring the velocities 
and energies of molecules, by using the properties of ideal gases. 
If T is the freezing point of water, and the Celsius temperature 
t is used instead of the absolute temperature T of the gas 
(J = 273 + 0, the above relation between the volume and the 
temperature of the gas may be written 

V = V i 1 +^yr) for p = constant. 

This is Charles' law, according to which the volume of the gas 
increases by 1/273 of its value at 0°C when the gas is heated by 1°. 
In deriving the equation of state for an ideal gas we have made 
no use of the fact that all its molecules are identical. This equa- 
tion is therefore valid also when the gas is a mixture of several 
different ideal gases — again a natural result of neglecting the 
interaction between molecules. It is only necessary to take N 
as the total number of gas molecules, i.e. the sum of the numbers 

of the various kinds of molecules: 7V = N 1 -fN 2 H , where 

N t is the number of molecules of the ith kind. Writing the equation 
of state of the gas as 

pV=N l kT + N 2 kT+- ■ ■ 

and noting that, if the whole volume V were occupied by molecules 
of the ith kind alone, the pressure p { would be such that 
p t V = N { kT, we conclude that 

p = Pi+p 2 +- , 

i.e. the pressure of a mixture of gases is equal to the sum of the 
pressures which each individual gas would exert if it alone filled 
the volume (Dalton's law). The pressures p u p 2 ,... are called the 
partial pressures of the respective gases. 

§54. An ideal gas in an external field 

Let us consider an ideal gas in a force field, for example the 
field of gravity. Since external forces then act on the gas mole- 
cules, the gas pressure will not be the same everywhere, but will 
vary from point to point. 



158 HEAT [VII 

For simplicity, we shall take the case where the field forces 
are in a fixed direction, which we choose as the z axis. We con- 
sider two unit areas perpendicular to the z axis and at a distance 
dz apart. If the gas pressures on the two areas are p and p + dp, 
the pressure difference dp must clearly be equal to the total force 
on the gas particles in a parallelepiped of unit base and height 
dz. This force is Fndz, where n is the density of molecules (i.e. 
the number of molecules per unit volume) and F the force on one 
molecule at a point with coordinate z. Hence 

dp = nF dz. 

The force F is related to the potential energy U(z) of a molecule 
by F = —dUldz, so that 

dp = —ndz . dU/dz = —ndU. 

Since the gas is assumed ideal, pV = NkT, and by using the 
relation NfV = n we can write this as p = nkT. We shall suppose 
that the gas temperature is the same at every point. Then 

dp — kTdn. 

Equating this to the above expression dp = —ndU, we find 

dnjn = dlog e n = —dU/kT, 
whence 

log c n = —U/kT + constant 
and finally 



n = n n e 



-UlkT 



where n is a constant which is evidently the density of molecules 
at a point where U = 0. 

The formula just derived which relates the variation in density 
of the gas to the potential energy of its molecules is called 
Boltzmann' s formula. The pressure differs from the density by a 



§54] AN IDEAL GAS IN AN EXTERNAL FIELD 159 

constant factor kT, and so a similar equation is valid for the 



pressure: 

p = p e~ ulkT . 



In the field of gravity near the Earth's surface, the potential 
energy of a molecule at height z is U = mgz, where m is the mass 
of a molecule. Thus, if the temperature of the gas is regarded as 
independent of height, the pressure p at height z is related to the 
pressure p on the Earth's surface by 

p = p e- maslkT . 

This is called the barometric formula ; it may be more conveniently 
written in the form 

p = p e~ >L9ZIRT , . 

where /x is the molecular weight of the gas and R the gas constant. 

This formula can also be applied to a mixture of gases. Since 
there is practically no interaction between the molecules of 
ideal gases, each gas may be treated separately, i.e. a similar 
formula is applicable to the partial pressure of each gas. 

The greater the molecular weight of a gas, the more rapidly 
its pressure decreases with increasing height. The atmosphere 
therefore contains an increasing proportion of light gases with 
increasing height; the content of oxygen, for example, decreases 
more rapidly than that of nitrogen. 

It should be remembered, however, that the applicability of the 
barometric formula to the real atmosphere is very limited, since 
the atmosphere is not in fact in thermal equilibrium and its 
temperature varies with height. 

An interesting conclusion can be drawn from Boltzmann's 
formula if we attempt to apply it to the atmosphere at all distances 
from the Earth. At very large distances Trom the Earth's surface, 
U must be taken not as mgz but as the exact value of the potential 
energy of a particle: 

U = —GMmfr, 
where G is the gravitational constant, M the Earth's mass and 



160 HEAT [VII 

r the distance from the centre of the Earth (see §22). Substituting 
this energy in Boltzmann's formula gives the following expression 
for the gas density: 

n = na> e GMm ' kTr , 

where now «„ denotes the gas density where U = (i.e. at an 
infinite distance from the Earth). Putting r here equal to the 
Earth's radius R, we find a relation between the densities of the 
atmosphere at the Earth's surface {n E ) and at infinity («<»): 

„ _ „ 0-GMmlkTR 

According to this formula, the density of the atmosphere at an 
infinite distance from the Earth should be non-zero. This con- 
clusion is absurd, however, since the atmosphere originates from 
the Earth, and a finite quantity of gas cannot be spread over an 
infinite volume with a density which is nowhere zero. The con- 
clusion is reached because we have tacitly assumed that the 
atmosphere is in a state of thermal equilibrium, which does not 
in fact exist. This result shows, however, that a gravitational 
field cannot retain a gas in a state of equilibrium, and the atmo- 
sphere should therefore be steadily dissipated into space. For 
the Earth this dissipation is extremely slow, and in its whole time 
of existence the Earth has not lost an appreciable fraction of its 
atmosphere. For the Moon, however, with its much weaker field 
of gravity, the atmosphere has been lost much more quickly, and 
in consequence the Moon now has no atmosphere. 

§55. The Maxwellian distribution 

The thermal velocity v T is a certain average property of the 
thermal motion of particles. In reality, different molecules move 
with different velocities and we may ask what is the velocity 
distribution of the molecules, that is, how many (on average) of 
the molecules in the body have a particular velocity? 

We shall derive the answer to this question for an ideal gas in 
thermal equilibrium. To do so, let us consider a column of gas in 
a uniform field of gravity, and first examine the distribution of 
molecules with respect to the values of only one velocity com- 
ponent, the vertical component v z . Let nf{v z )dv z denote the 



§55] THE MAXWELLIAN DISTRIBUTION 161 

number of molecules per unit volume of the gas for which the 
value of this component lies in an infinitesimal interval between 
v z and v z + dv z . Here n is the total number of molecules in the 
volume considered, and so f(v z ) determines the fraction of 
molecules having a particular value of v z . 

Let us consider molecules with velocities in the interval dv z 
which are in a layer of gas at height z and of infinitesimal thickness 
dz. The volume of this layer is equal to dz if the cross-section of 
the gas column is of unit area, and the number of such molecules 
is therefore n(z)f(v z )dv z dz, where n(z) is the density of gas 
molecules at height z. These molecules move as free particles 
(since the collisions in an ideal gas may here be neglected) and 
subsequently reach a different layer of thickness dz' at height 
z', with velocities in the interval between some values v z and 
v z + dv z . Since the number of molecules is unchanged, we have 

n(z)f(v z )dv z dz = n(z')f(v z ')dv z 'dz'. 

For movement in a field of gravity, the horizontal velocity 
components v x , v y remain constant, and the change in v z is deter- 
mined by the law of conservation of energy, according to which 

?mv z 2 + mgz = %mv z ' 2 4- mgz' . 

Differentiating this equation (for given constant values of z 
and z') we obtain 

v t dv z = v z ' dv z ' 

as the relation between dv z and dv z ' , the ranges of values of the 
vertical velocities of the molecules considered, at heights z and 
z'. The thicknesses dz and dz' of the layers are related by 

dz/v z = dz'/v g '; 

this simply expresses the fact that in a time dt = dzlv z during 
which a molecule crosses a layer dz at height z, it will travel a 
distance dz' = v 2 'dt at height z'. Multiplication of the two 
equations above gives 

dv z dz = dv'dz'. 



162 HEAT [VII 

In the condition of constant number of molecules shown previous- 
ly, therefore, the differentials on the two sides of the equation 
cancel, leaving 

n{z)f{v z ) = n(z')f{v z '). 

The barometric formula states that 

n{z)ln(z') = e~ m(,hlkT 

(where h = z — z' is the difference in height), and hence 

f(v z ')=f{v z )e- m ^ kT . 

Thus the required distribution function must be multiplied by 
e -mghikT w h en ± mV; z i s replaced by \mv z 2 = \mv z 2 + mgh. The 
only function having this property is the exponential function 

f(v z ) = constant X e~ mv ^ l2kT . 

[It should be noted that the acceleration due to gravity does not 
appear in this formula. This is as it should be, since the mechan- 
ism of establishment of the velocity distribution of the gas 
molecules consists in collisions between molecules and does 
not depend on the external field. In the foregoing derivation the 
field served only the auxiliary purpose of relating the velocity 
distribution to the already known Boltzmann's formula.] 

We have found the equilibrium distribution of molecules with 
respect to one component of the velocity. The fraction of 
molecules having given values of all three velocity components 
simultaneously is evidently obtained by multiplying together 
the fractions of molecules having given values of each component 
separately. Thus the complete distribution function is 

f(v x , V y , V z ) = COnStant X e -™ a *l2kT e -mv i ?l2Kr e -mv*l2kT m 

Adding the exponents and using the fact that the sum v 2 + v u 2 + 
v z 2 is v 2 , the square of the magnitude of the velocity, we have 
finally 

/= constant x e - mv2t2kT . 



§55] THE MAXWELLIAN DISTRIBUTION 163 

The number dN of gas molecules whose velocity components 
lie in the intervals between v x , v y , v g and v x + dv x , v y + dv y , v z + 
dvg is therefore 

dN = constant X e- mv2 ' 2kT dv x dv y dv z ; 

the constant coefficient is determined by the condition that the 
total number of molecules with all possible velocities is equal 
to the given number N of molecules in the gas, but its value will 
not be written out here. The formula derived above is called 
the Maxwellian distribution formula. 

The analogy between this formula and Boltzmann's formula 
for the gas density distribution in space in an external field 
should be noted: in each formula we have an exponential expres- 
sion of the form e~ elkT , where e is the energy of a molecule (the 
kinetic energy \mv 2 for the velocity distribution, and the potential 
energy U(x,y,z) in the external field for the distribution in space). 
This exponential expression is often called a Boltzmann factor. 

If the three components v x , v y , v z are given, both the magnitude 
and the direction of the velocity of the molecule are determined. 
But the distribution of molecules with respect to the direction 
of the velocity is simply a uniform distribution, with equal 
numbers, on average, travelling in every direction. [This follows 
from the fact that the Maxwellian distribution involves only the 
absolute magnitude v of the velocity, but it is also evident a 
priori: if there existed some preferred direction of motion of 
the molecules in the gas, this would mean that the gas was not 
at rest but was moving in that direction.] 

The Maxwellian formula can be transformed so as to give 
directly the distribution of gas molecules with respect to absolute 
magnitude of velocity regardless of direction. For this purpose 
we must take the total number of molecules with various values 
of the velocity components v x , v y , v z but a given value of v 2 = 
v x 2 + v y 2 + v 2 . This is easily done by using the following geo- 
metrical analogy. If we use a coordinate system with the values 
of v x , v y , v z plotted along the axes, the product dv x dv y dv z will 
be the volume of an infinitesimal parallelepiped with edges dv x , 
dv y , dv z . We must sum over all volume elements at a fixed 
distance from the origin (since v is clearly the length of the 
"radius vector" in these coordinates). These volumes occupy 



164 



HEAT 



[VII 



a spherical shell between two spheres of radii v and v + dv. The 
volume of the shell is equal to the area 4ttv 2 of the spherical 
surface multiplied by the thickness dv of the shell. 

Thus, replacing the product dv x dv y dv z in the Maxwellian 
distribution formula by Airv 2 dv, we find the number of molecules 
with velocities in the interval from v to v + dv: 



dN = constant X e 



-mv 2 l2kT 



v 2 dv. 



The coefficient of dv in this formula is the number of molecules 
per unit interval of velocity. As a function of v it has the form 
shown in Fig. 73. It is zero when v = 0, reaches a maximum for 
a value v , and tends very rapidly to zero as the velocity increases 
further. The maximum on the curve corresponds to the value 
v =V(2kTlm), which is slightly less than the thermal velocity 
v T defined in §50. 




Fig. 73. 

Since different molecules have different velocities, it makes 
a difference, in determining the mean properties, which quantity 
is averaged. For example, the mean value ~v_ of the velocity 
itself is not the same as the velocity v T = Vu 2 (which is often 
called also the root-mean-square velocity, in order to stress its 
origin). From the Maxwellian distribution it can be shown that 
v = 0-92u r . 

The Maxwellian distribution has been derived here for a 
monatomic gas, but it can in fact be deduced from much more 
general theoretical arguments, and is a universal result. It is 
valid for the thermal motion of molecules and atoms in all 
bodies, but it is based on classical mechanics and its validity 
is limited by quantum effects in the same way as the applicability 
of classical mechanics in general to thermal motion. 



§55] 



THE MAXWELLIAN DISTRIBUTION 



165 



The velocity distribution in thermal motion can be studied 
by various methods using molecular beams. These are obtained 
by allowing molecules to evaporate into an evacuated vessel 
from a substance heated in a special type of furnace. The vessel 
is evacuated to such an extent that molecules move in it almost 
without collisions. 

One such method is based on the idea of a mechanical velocity 
selector, which works in the following way. Two circular discs 
with radial slots at an angle a to each other rotate on a common 
axis at a distance / apart in an evacuated vessel (Fig. 74). A 
molecular beam from the furnace F passes through the diaphragm 
D to the discs. A molecule which passes through the slot in the 
first disc with velocity v will reach the second disc after a time 
t = II v. In this time the disc turns through an angle Qt = Cll/v, 
where CI is the angular velocity of rotation. Thus only molecules 
whose velocity is such that Q,llv = a will pass through the slot 
in the second disc and leave a trace on the screen S. By varying 
the speed of rotation of the discs and measuring the density of 
the deposit on the screen we can find the relative numbers of 
particles with various velocities. 




■e- 




c-- 



Fig. 74. 



3 
Fig. 75. 



h 
I 4 



The Maxwellian distribution has also been tested experi- 
mentally by observing the deviation of a molecular beam under 
gravity. Atoms of caesium heated in the furnace 1 (Fig. 75) and 
emerging from an aperture in it enter an evacuated vessel. A 
narrow beam selected by the diaphragms 2 and 3 is deflected 
downwards by gravity and is collected by a detector in the form 
of a heated thin horizontal tungsten wire 4 which can be placed 
at various distances h below the axis of the apparatus; the 
caesium atoms which strike the wire leave it as positive ions 
which are collected by a negatively charged plate. The deflection 
h of an atom depends on its velocity v; in the experiments, this 
deflection was some tenths of a millimetre with a beam path 



166 



HEAT 



[VII 



length of 2 metres. By measuring the beam intensity for various 
values of h we can find the velocity distribution of the atoms in 
the beam. 

§56. Work and quantity of heat 

When a body expands, it moves the surrounding bodies, i.e. 
does work on them Let us consider, for example, a gas beneath 
a piston in a cylindrical vessel. If the gas expands and moves 
the piston an infinitesimal distance dh, it does work dA on the 
piston, where dA = F dh and F is the force exerted by the gas 
on the piston. But, by definition, F = pS, where p is the gas 
pressure and S the area of the piston. Hence dA = pSdh, and 
since Sdh is the increase dV in the volume of the gas we have 
finally 

dA=pdV. 

This simple and important formula determines the work done 
in an infinitesimal change in the volume of a body We see that 
this work depends only on the pressure and the total change in 
volume, and not on the shape of the body. [To avoid misunder- 
standing it should be mentioned at once that this assertion does 
not apply to solids ; see § 1 1 .] 

The work dA is positive when the body expands (dV > 0), and 
the body does work on the surrounding medium. When the body 
is compressed (dV < 0), on the other hand, work is done on it 
by the surrounding bodies, and with our definition of dA this 
corresponds to negative work. 

The work done in a given process can be represented by a 
geometrical analogy if the process is shown graphically as a 
curve in the coordinates p and V. For example, let the change in 




§56] 



WORK AND QUANTITY OF HEAT 



167 



pressure of a gas as it expands be shown by the curve 12 in Fig. 
76. When the volume increases by dV, the work done by the 
gas is pdV, i.e. the area of the infinitely narrow rectangle re- 
presented by the hatched area in the diagram. The total work 
done by the gas in expanding from volume V x to V 2 therefore 
consists of the elements of work dA whose sum is represented 
by the area \2V 2 V X below the curve and between the two extreme 
vertical lines. Thus the area in the diagram gives at once the 
work done by the body in the process considered. 




Fig. 77. 



One frequently encounters cyclic processes, i.e. those in which 
the body finally returns to its original state. For example, let a 
gas be subjected to the process shown by the closed curve lalbl 
in Fig. 77. On the curve 1«2 the gas expands and does work 
represented by the area under that curve; on the curve 2M the 
gas is compressed, and the work done is therefore negative and 
equal in magnitude to the area under the curve 2bl. The total 
work done by the gas is consequently equal to the difference of 
these areas, i.e. is represented by the hatched area in Fig. 77 
lying within the closed curve. 

The total work A done by the body in expansion from volume 
Vi to V 2 is given by a particularly simple expression when the 
process occurs at constant pressure. In this case we clearly have 

A=p(V 2 -V 1 ). 

We may also determine the work done in an isothermal 
expansion of an ideal gas. For one gram-molecule of gas the 
pressure p = RT/V; hence 

dA = p dV = (RT/V)dV =RT d log e F; 



168 HEAT LVH 

since the temperature remains constant, we can write dA = 
d(RT logeV). Hence it follows that the work A is equal to the 
difference between the values of RT log e V at the end and the 
beginning of the process, i.e. 

A = RT log e (VJV t ). 

If the body gains no energy from external sources, the work 
done in expansion is done at the expense of its internal energy. 
This energy, which we denote by E, includes the kinetic energy 
of the thermal motion of the atoms of the substance and the 
potential energy of their mutual interaction. 

However, the change in the internal energy of the body in a 
given process is not in general equal to the work done. The 
reason is that the body may also gain (or lose) energy by direct 
transfer from other bodies without doing mechanical work. The 
energy thus gained is called the quantity of heat gained by the 
body; we shall regard it as positive if the body gains heat and 
negative if it loses heat. 

Thus the infinitesimal change in the internal energy of the 
body consists of two parts: an increase due to the quantity of 
heat gained by the body (which we denote by dQ) and a decrease 
due to the work dA done by the body. Hence we have 

dE = dQ-pdV. 

This important relation expresses the law of conservation of 
energy for thermal processes and is called in this connection 
theirs/ law of thermodynamics. 

It must be emphasised that the work and the quantity of heat 
depend not only on the initial and final states of the body but also 
on the path along which the change in the state of the body takes 
place. For this reason we cannot speak of the "quantity of heat 
contained in a body" and regard the amount of heat concerned 
in the process as the difference of this quantity in the final and 
initial states. The fact that such a quantity has no meaning is 
especially clear if we consider a cyclic process, where the body 
returns to its initial state but the total amount of heat gained 
(or lost) is certainly not zero. 

Only the internal energy E is what is called a. function of the 
state: in any given state, the body has a definite energy. The 



§56] WORK AND QUANTITY OF HEAT 169 

total change in the energy of a body during a process is therefore 
a quantity depending only on the final and initial states, namely 
the difference E 2 — E x between the energies in these states. The 
separation of this change into a quantity of heat Q and an amount 
of work A is not unique, but depends on the path taken in going 
from the initial to the final state. In particular, in a cyclic process 
the total change in energy is zero; the quantity of heat Q gained 
by the body and the work/4 done by it are not zero, but Q = A. 

In thermal measurements a special unit of energy, the calorie 
(cal), was used until recently. The definition of this unit as the 
quantity of heat needed to heat 1 g of water by 1° is insufficiently 
exact, since the specific heat of water depends slightly on the 
temperature. In consequence, various definitions of the calorie 
existed which differed somewhat in value. The relation between 
the calorie and the joule is approximately 

leal = 4- 18 J. 

If the temperature of one gram-molecule of a substance is 
raised by dT when it gains a quantity of heat dQ, the ratio 

C = dQldT 

is called the specific heat of the substance. This definition, 
however, is inadequate by itself, since the quantity of heat 
necessary depends not only on the change in temperature but 
also on the other conditions under which the heating takes place: 
it is necessary to state how other properties of the substance be- 
sides the temperature are affected. Because of this indefiniteness, 
various definitions of the specific heat are possible. 

The most usual in physics are the specific heat at constant 
volume C v and the specific heat at constant pressure C p , which 
give the quantities of heat when the substance is heated under 
conditions such that its volume and pressure respectively remain 
constant. 

If the volume remains constant, then dV = and dQ = dE, i.e. 
all the heat is used to increase the internal energy. We can 
therefore write 

C F = (dEldT) v . 



170 HEAT [VII 

The suffix V to the derivative signifies that the differentiation is 
to be taken for a constant value of V. This indication is necessary, 
since the energy of a body depends, in general, not only on the 
temperature but also on other quantities describing the state of 
the body, and the result of the differentiation therefore depends 
on which of these quantities is assumed constant. 

If the pressure remains constant on heating, then heat is used 
not only to increase the internal energy but also to do work. In 
this case the quantity of heat may be written in the form 

dQ = dE+pdV=d(E+pV), 

since p = constant. We see that the quantity of heat is equal to 
the change in the quantity 

W = E+pV. 

This is called the enthalpy, heat function or heat content; like 
the energy, it is a definite function of the state of the body. Thus 
the specific heat at constant pressure may be calculated as the 
derivative 

C p = (dW/dT) p . 

The specific heat C p is always greater than C v : 

At first sight it might appear that this inequality is due simply 
to the work which must be done by a body in expanding on 
heating at constant pressure. This is not so, however; the in- 
equality applies also to the few substances which contract on 
heating, as well as to those which expand. It is in fact a con- 
sequence of a very general theorem of thermodynamics: an 
external interaction which removes a body from a state of 
thermal equilibrium brings about processes in it which, as it 
were, try to reduce the effect of this interaction. For example, 
heating a body brings about processes which absorb heat, 
whereas cooling brings about processes in which heat is evolved. 
This is called Le Chatelier's principle. 



§57] THE SPECIFIC HEAT OF GASES 171 

Let us imagine that a body in equilibrium with an external 
medium receives a quantity of heat in such a way that its volume 
remains unchanged and its temperature increases by an amount 
(ATV The pressure of the body will also be changed, and the 
equilibrium condition, according to which this pressure must be 
equal to that of the surrounding medium, will no longer be 
satisfied. According to Le Chatelier's principle, the restoration 
of equilibrium, which would restore the original pressure, must 
be accompanied by cooling. In other words, the change (AT) P 
in the temperature of the body at constant pressure is less than 
the change (AT) K at constant volume (for a given quantity of 
heat gained by the body). This means that, for a given change in 
temperature, more heat is necessary at constant pressure than 
at constant volume. 

In what follows we shall several times make use of Le Chate- 
lier's principle to decide the direction in which a quantity changes 
when another quantity is varied. 

§57. The specific heat of gases 

Since the molecules of an ideal gas are assumed not to interact 
with one another, the change in their mean distance apart when 
the volume of the gas varies cannot affect its internal energy. 
In other words, the internal energy of an ideal gas is a function 
only of its temperature, and not of its volume or pressure. Hence 
the specific heat C v = dE/dT of the gas also depends only on 
the temperature. 

The same is true of the specific heat C p = dW/dT, and there is 
a very simple relation between the two specific heats of the gas. 
From the equation of state pV = RT, the enthalpy of one mole 
of gas is related to its internal energy by 

W = E + pV=E + RT. 

Differentiating this expression with respect to temperature, we 
obtain 

i.e. the difference of the molar specific heats of the gas, C P ~C V , 
is equal to the gas constant R = 8-3 J/deg.mole = 2 cal/deg.mole. 



172 HEAT [VII 

It is easy to find the specific heat of a monatomic gas (such as 
the noble gases). In this case the internal energy of the gas is 
simply the sum of the kinetic energies of the translational motion 
of the particles. Since, by the definition of temperature, the mean 
kinetic energy of one particle is f kT, the internal energy of one 
mole of gas is 

E = %N-JcT = %RT. 

The specific heats are therefore 

C v = %R = 12-5 J/deg.mole, 
C P = %R = 20-8 J/deg.mole. 

These values are quite independent of temperature. 

We shall see later that in many processes an important property 
of the gas is the ratio of the specific heats C p and C v , usually 
denoted by y: 

y = C P IC V . 

For monatomic gases 

7=5/3 = 1-67. 

The specific heat of diatomic and polyatomic gases is more 
complicated than that of monatomic gases. Their internal energy 
consists of the kinetic energies of translation and rotation of 
the molecules and the energy of the atoms vibrating within the 
molecule. Thus each of these three types of motion makes a 
certain contribution to the specific heat of the gas. 

Here we may return to the definition of temperature given in 
§50. Since a molecule has three degrees of freedom in its trans- 
lational motion, we can say that each of them corresponds to a 
mean kinetic energy ikT. According to classical mechanics, the 
same result would be obtained for every degree of freedom of the 
molecule, whether for translational motion, rotation, or vibration 
of the atoms within it. We know also that in the vibrational 
motion the mean value of the potential energy is equal to the 
mean value of the kinetic energy. Thus, according to classical 



§57] THE SPECIFIC HEAT OF GASES 173 

mechanics, the thermal potential energy of each degree of 
freedom of the vibration of atoms within the molecule would 
also be $kT. Thus we find that any gas should have a constant 
specific heat independent of temperature and determined entirely 
by the number of degrees of freedom of the molecule (and 
therefore by the number of atoms in it). 

In reality, however, the vibrational motion of the atoms in 
the molecule affects the specific heat of the gas only at sufficiently 
high temperatures. The reason is that this motion remains of the 
nature of "zero-point vibrations", not only at low temperatures 
but also at comparatively high temperatures, on account of 
the comparatively large energy of these vibrations. The "zero- 
point energy", by its nature, is independent of temperature, and 
therefore does not affect the specific heat. For example, in the 
molecules of diatomic gases (nitrogen, oxygen, hydrogen etc.), 
the vibrations of the atoms within the molecules are fully "in- 
cluded" in the motion only at temperatures of the order of 
thousands of degrees; at lower temperatures their contribution 
to the specific heat decreases rapidly and is practically zero 
even at room temperature. 

The zero-point energy of rotation of molecules is very small, 
and thus classical mechanics is very soon applicable to this 
motion: at temperatures of a few degrees Kelvin for diatomic 
molecules, with the exception of the lightest gas, hydrogen, for 
which a temperature of about 80°K is necessary. 

In the neighbourhood of room temperature, the specific heat 
of diatomic gases is therefore due only to the translational and 
rotational motion of the molecules and is very close to its 
theoretical constant value (in classical mechanics) 

CV = f R = 20-8 J/deg.mole, 
C p = iR = 291 J/deg.mole. 

The ratio of specific heats y = 7/5 = 1 -4. 

We may note that in the "quantum region" the mean energies 
of the thermal rotational and vibrational motions, and therefore 
the specific heat of the gas, depend not only on the temperature 
but also on the "individual" properties (moments of inertia and 
vibrational frequencies) of the molecule. [It is for this reason 



174 HEAT [VII 

that these energies, unlike the energy of the translational motion, 
cannot be used for a direct definition of temperature.] 

The specific heat of polyatomic gases is even more complicated. 
The atoms in a polyatomic molecule can execute oscillations of 
various types with various zero-point energies. As the temper- 
ature rises, these oscillations are successively "included" in 
the thermal motion, and the specific heat of the gas increases 
accordingly. It may happen, however, that the inclusion of all 
the oscillations is never achieved, since the molecules may 
disintegrate at high temperatures. 

It may again be recalled that the whole of the above discussion 
is for the case of an ideal gas. At high pressures, when the 
properties of the gas become appreciably different from those of 
an ideal gas, its specific heat is also changed, because of the 
contribution to the internal energy arising from the interaction 
between the molecules. 

§58. Solids and liquids 

The simplicity of the thermal properties of an ideal gas, which 
allows a general equation of state for all gases to be derived, is 
due to the fact that the interaction between molecules in the 
gas is unimportant. In solids and liquids, the interaction between 
the molecules is of primary importance; the thermal properties 
of these substances therefore differ considerably, and it is 
impossible to establish any general equation of state. 

Solids and liquids, unlike gases, are not readily compressed. 
The compressibility of a substance is usually defined as 

= -K*L\ • 
K V\dp ) T > 

the derivative of the volume with respect to the pressure is taken 
at constant temperature, i.e. describes a process of isothermal 
compression. This coefficient is negative, i e. the volume decreases 
when the pressure increases, and the minus sign is used in order 
to make the compressibility a positive quantity. The dimensions 
of k are evidently the reciprocal of those of pressure. 

As examples, we may give the values of the compressibility 
per bar for various liquids at room temperature and atmospheric 
pressure: 



§58] SOLIDS AND LIQUIDS 175 

Mercury 0-4 x lO" 5 bar -1 Alcohol 7-6 x 10~ 5 bar -1 
Water 4-9xl0~ 5 Ether 14-5 xl0~ 5 

The compressibilities of most solids are even smaller: 



Diamond 


0- 16 x lumbar" 1 


Aluminium 


1 -4 x lumbar" 1 


Iron 


0-61 x 10- 6 


Glass 


2-7 x 10- 6 


Copper 


0-76 x 10- 6 


Caesium 


62 x 10- 6 



For comparison, let us find the compressibility of a gas. In iso- 
thermal compression, the volume of a gas decreases in inverse 
proportion to the pressure: V= RT/p. Substituting this expres- 
sion in the foregoing definition of the compressibility k, we have 
after the differentiation 

k = Up. 

At a pressure of 1 bar, the compressibility of the gas is 1 bar -1 . 

Another quantity used to describe the thermal properties of 
solids and liquids is the coefficient of thermal expansion defined 
as 

the suffix p to the derivative means that the body is heated at 
constant pressure. 

The majority of bodies expand on heating, and the coefficient 
a is positive. This is to be expected, since the greater thermal 
motion tends to move the molecules apart. Nevertheless, there 
are exceptions to this rule. For example, water contracts on 
heating in the range from to 4°C. Liquid helium also contracts 
on heating at temperatures below 219°K (helium II; see §74). 

As examples, we may give the coefficients of thermal expan- 
sion of various liquids at room temperature: 

Mercury 1 -8 X 10~ 4 deg" 1 Alcohol 10-8 x 10~ 4 deg" 1 
Water 2-1 x 10~ 4 Ether 16-3 x 10~ 4 

[For comparison, the coefficient of thermal expansion of gases, 



176 HEAT [VII 

obtained by substituting V=RT/p in the definition of a, is 
a = l/T; for T = 293°K, a = 3-4 X 10~ 3 deg" 1 .] 
The coefficient of thermal expansion of solids is still smaller: 

Iron 3-5xl0- 5 deg- x 
Copper 5-0xl0~ 5 
Glass 2-4to30xl0- 5 

Invar (an alloy of 64% iron and 36% nickel) and fused quartz 
have especially small values of a (3 x 10 -6 and 1-2 x 10 -6 respec- 
tively). These substances are widely used in making parts of 
instruments in which it is desirable to avoid dimensional changes 
when the temperature varies. 

It has been mentioned in §45 that the thermal expansion of 
crystals (other than cubic) occurs differently in different direc- 
tions. This difference may be very considerable. For example, 
in the thermal expansion of a crystal of zinc, the linear dimension 
in the direction of the hexagonal axis increases 4-5 times faster 
than those in the directions perpendicular to this axis. 

The specific heat of solids and liquids, like that of gases, 
usually increases with temperature. The specific heat of a solid 
depends on the energy of atoms executing small thermal oscilla- 
tions about their equilibrium positions. When the temperature 
rises, this specific heat tends to a certain limit corresponding to 
the state where the oscillations of the atoms can be treated on 
the basis of classical mechanics. Since the motion of the atoms is 
entirely oscillatory, a mean energy kT must correspond to each 
of its three degrees of freedom: \kJ from the mean kinetic energy 
and \kT from the mean potential energy (as described in §57). 
The total mean energy per atom in a solid would then be 3kT. 

This limit, however, is never reached for compounds of any 
complexity, since the substance melts or decomposes before this 
occurs. At ordinary temperatures the limiting value of the 
specific heat is reached for many elements, so that the specific 
heat of one gram-atom of a solid element is approximately 

C = 3R = 25 J/deg.mole = 6 cal/deg.mole; 

this is sometimes called Dulong and Petit' s law. 

In discussing the specific heat of a solid we deliberately do not 
distinguish between the specific heats at constant pressure and 



§58] SOLIDS AND LIQUIDS 177 

at constant volume. The measured specific heats are usually 
those at constant pressure, but in solids the difference between 
C p and C v is very small (e.g. for iron C P IC V = 1-02). This is 
because of the smallness of the coefficient of thermal expansion 
for solids: there is a general relation between the difference of 
specific heats for any body, the coefficient of thermal expansion 
a and the compressibility k: 

c p —Cv= Ta 2 lpK, 

where p is the density of the substance and c p and c v the specific 
heats per gram. Thus we see that the difference c p — c v is propor- 
tional to the square of the coefficient a. 

As the temperature decreases, the specific heat of a solid also 
decreases and tends to zero at absolute zero. This is a conse- 
quence of a remarkable general theorem (called Nernst's theorem), 
according to which, at sufficiently low temperatures, any quantity 
representing a property of a solid or liquid becomes independent 
of temperature. In particular, as absolute zero is approached, the 
energy and enthalpy of a body no longer depend on the 
temperature; the specific heats c p and c v , which are the deriva- 
tives of these quantities with respect to temperature, therefore 
tend to zero. 

It also follows from Nernst's theorem that, as T — » 0, the 
coefficient of thermal expansion tends to zero, since the volume 
of the body ceases to depend on the temperature. 



CHAPTER VIII 

THERMAL PROCESSES 



§59. Adiabatic processes 

Let us now consider some simple thermal processes. A very 
simple process is the expansion of a gas into a vacuum: the gas 
is initially in a part of a vessel separated from the rest of the vessel 
by a partition, and then an opening is made in the partition and 
the gas fills the whole vessel. Since the gas does no work in such 
an expansion, its energy remains constant, i.e. the energy E t of 
the gas before the expansion is equal to its energy E 2 after the 
expansion: 

Ei = E 2 . 

For an ideal gas the energy depends, as we know, only on the 
temperature; thus, since the energy is constant, it follows that 
the temperature of an ideal gas remains constant when it expands 
into a vacuum. However, the temperature of gases which are 
not nearly ideal changes on expansion into a vacuum. 

There is another process of expansion of a gas called an 
adiabatic process, which differs very greatly from expansion 
into a vacuum. Adiabatic processes are of great importance, and 
will now be considered in detail. 

The typical feature of an adiabatic process is that the gas 
remains continuously under an external pressure equal to the 
pressure of the gas itself. Another condition for an adiabatic 
process is that throughout the process the gas remains thermally 
isolated from the external medium, i.e. does not gain or lose 
heat. 

It is simplest to imagine the adiabatic expansion (or compres- 
sion) of a gas in a thermally isolated cylindrical vessel with a 
piston. When the piston is moved out sufficiently slowly, the gas 
expands behind it and at every instant has a pressure correspond- 

178 



§59] ADIABATIC PROCESSES 179 

ing to the total volume which it then occupies. Here "sufficiently 
slowly" means, therefore, so slowly that the gas is able to estab- 
lish thermal equilibrium corresponding to every instantaneous 
position of the piston. If, on the other hand, the piston is moved 
out too rapidly, the gas will not be able to follow it, and a region 
of reduced pressure will exist beneath the piston, into which 
the remaining gas will expand; similarly, if the piston is moved in 
too rapidly, a region of increased pressure will exist. Such 
processes would not be adiabatic. 

In practice, this condition of slowness is very easily fulfilled 
in the case considered. Analysis shows that the condition would 
not be fulfilled only if the rate of movement of the piston were 
comparable with the velocity of sound in the gas. Thus, in the 
practical carrying out of an adiabatic expansion, the principal 
condition is that of thermal isolation, which requires that the 
process should be "sufficiently fast": the gas must not be able to 
exchange heat with the external medium during the process. It 
is clear that this condition is entirely compatible with the condi- 
tion of "sufficient slowness" stated above; it depends on the 
thoroughness of the thermal isolation of the vessel and may be 
said to be of secondary importance and unrelated to the actual 
nature of the process. For this reason an adiabatic process is 
regarded in physics as one which primarily satisfies the condition 
of "sufficient slowness", the latter being fundamental. We shall 
return to a discussion of this condition in §62. 

In an adiabatic process we can not say that the internal energy 
of the gas itself remains constant, since the gas does work when 
it expands (or work is done on it when it is compressed). The 
general equation of an adiabatic process is obtained by putting 
the quantity of heat dQ equal to zero in the relation dQ = 
dE + pdV, in accordance with the condition of thermal isolation. 
Thus an infinitesimal change in the state of a body in an adiabatic 
process is described by the equation 

dE+pdV=Q. 

Let us apply this equation to the adiabatic expansion (or 
compression) of an ideal gas; for simplicity, all quantities will 
refer to one mole. The energy of an ideal gas is a function only 
of its temperature, and the derivative dE/dT is the specific heat 



180 THERMAL PROCESSES [VIII 

CV; in the equation of the adiabatic process, we can therefore 
replace dE by C v dT: 

C v dT+pdV=0. 

Substituting p = RT/V and dividing the equation by T, we obtain 
the relation 

C v dT/T + RdV/V=0. 

Let us assume further that the specific heat of the gas is 
constant in the temperature range considered; for monatomic 
gases this is always true, and for diatomic gases it is true over a 
wide range of temperatures. Then the above relation may be 
written 



d(C v \og e T + R\og e V) = 0, 



whence 



C v log e T + R \og e V = constant 

or, in power form, 

jcvyR = constant. 

Finally, since for an ideal gas C p -C v = R, the \\C V power of 
this equation may be written 

TVy- 1 = constant, 

where y = C p /C v . 

We see that in an adiabatic process the temperature and volume 
of an ideal gas vary in such a way that the product TV y ~ r remains 
constant. Since y is always greater than unity, y—\ > 0, and 
therefore an adiabatic expansion is accompanied by a cooling 
of the gas, and an adiabatic compression by heating. 

On combining the above equation with the formula pV = RT 
we can derive a similar relation between the temperature and the 
pressure in an adiabatic process: 



§59] 



and the relation 



ADIABATIC PROCESSES 

jp-{y-\)iy — constant, 



pV y = constant 



181 



between the pressure and the volume; this last relation is called 
the equation of Poisson's adiabatic. 

In isothermal expansion of a gas, its pressure decreases in 
inverse proportion to the volume V. In adiabatic expansion, we 
see that the pressure decreases in inverse proportion to V y , i.e. 
more rapidly (since y > 1). If these processes are represented 
graphically by plotting p against V as two curves, an isothermal 
and an adiabatic, intersecting at a point p , V which represents 
the initial state of the gas, then the adiabatic curve will be steeper 
than the isothermal (Fig. 78). 



p* 




Isothermal 



Adiabatic 



Fig. 78. 

This property may be stated in another manner by considering 
the change in volume as a function of pressure (i.e. by turning 
Fig. 78 through 90°) and representing this relation by the com- 
pressibility k = —(l/V)dVfdp; see §58, where the isothermal case 
was considered. It is then easy to see that the adiabatic com- 
pressibility of a gas is less than its isothermal compressibility: 

Kad *"■- K\ s . 

This inequality, derived here for gases, is in fact valid for all 
bodies, and follows from Le Chatelier's principle. 



182 



THERMAL PROCESSES 



[VIII 



On the other hand, another property of adiabatic processes in 
a gas, the heating on compression, is not a universal property of 
adiabatic compression of all bodies. This is likewise seen from 
Le Chatelier's principle. If a body is compressed without gaining 
any heat (which itself would affect the temperature of the body), 
the temperature of the body will change so as to oppose the com- 
pression. For the great majority of bodies, which expand on 
heating, this means that the temperature will rise on adiabatic 
compression (and conversely will fall on expansion). But it is 
clear from this discussion that, if the volume of a body decreases 
on heating, an adiabatic compression of the body will be 
accompanied by cooling. 

§60. Joule-Kelvin processes 

Processes in which a gas or liquid passes steadily from one 
pressure to another without exchange of heat with the surround- 
ing medium are of considerable interest. By "steadily" we here 
mean that the two pressures remain constant throughout the 
process. 

Such a process is in general accompanied by a flow of gas 
(or liquid) with some velocity different from zero, but this 
velocity can be made very small by causing the gas to go from 
one pressure to the other through an obstruction which greatly 
impedes the flow, such as a porous partition or a small hole. 



m 



w 



p 



(a) 




Fig. 79. 



The steady passage of a thermally isolated gas from one pres- 
sure to another under conditions where the gas does not acquire 
any appreciable velocity is called a Joule-Kelvin process. This 
process may be diagrammatically represented by the passage of 
a gas in a cylindrical vessel through a porous partition P (Fig. 
79a, b), the pressures p x and p 2 on each side of the partition being 
maintained constant by pistons 1 and 2. 



§60] JOULE-KELVIN PROCESSES 183 

Let the gas initially occupy a volume V x between piston 1 and 
the partition P (Fig. 79a). Piston 1 is now moved in and piston 
2 moved out, keeping the pressures p x and/? 2 acting on the pistons 
unchanged. The gas, passing at a low velocity through the porous 
partition, will finally occupy a volume V 2 between the partition 
and piston 2, and will be at a pressure p 2 (Fig. 79b). 

Since in this process there is no exchange of heat with the 
surrounding medium, the work done by the pistons must be equal 
to the change in the internal energy of the gas. The gas pressures 
remain constant during the process, and therefore the work done 
by the piston 1 in displacing the gas from the volume V x is simply 
the product p x V x . The gas passing through the partition does 
work on the piston 2. Thus the total work done by the pistons on 
the gas is PiV 1 —p 2 V 2 , and this, as already stated, must be equal 
to the increase in the internal energy of the gas: 

p l V 1 -p 2 V 2 = E 2 -E u 

where E x and E 2 are the internal energies of a given quantity of 
the gas in the initial and final states. Hence 

E l +p 1 V l = E 2 +p 2 V 2 , 

or 

W, = W 2 , 

where W = E+pV is the enthalpy. Thus the enthalpy of the gas 
is conserved in a Joule-Kelvin process. 

For an ideal gas both the energy and the enthalpy depend only 
on the temperature. Thus the equality of the enthalpies implies 
that the temperatures are equal: if an ideal gas undergoes a 
Joule-Kelvin process, its temperature remains unchanged. 

In real gases the temperature changes in a Joule-Kelvin 
process, and may do so by a considerable amount. For example, 
when air at room temperature expands from 200 atm pressure to 
1 atm, it is cooled by about 40°. 

At sufficiently high temperatures, all real gases are heated by 
expansion in a Joule-Kelvin process, while at lower temperatures 
(and not too high pressures) they are cooled; there is therefore a 



184 THERMAL PROCESSES [VIII 

temperature (called the inversion point) above which the change 
in temperature in a Joule-Kelvin process is of opposite sign. 
The position of the inversion point depends on the pressure, and 
is different for different gases. For example, air is cooled in a 
Joule- Kelvin process at room temperature, but to achieve this 
effect in hydrogen it must first be cooled to about 200°K or below, 
and for helium a temperature of 40°K is necessary. 

The change in temperature in a Joule-Kelvin process is widely 
used in technology for the liquefaction of gases. The gas velocity 
is usually lowered by means of a narrow opening called an 
expansion valve. 

§61. Steady flow 

In a Joule-Kelvin process, the gas passes steadily from one 
pressure to the other, and its velocity is artificially made small 
by means of friction. However, the results obtained by consider- 
ing this process are easily generalised to the case of any steady 
thermally isolated flow of gas (or liquid) with non-zero velocity. 
The only difference is that the kinetic energy of the flowing gas 
can not now be neglected. The work done on the gas increases its 
energy, which now includes the kinetic energy of its motion as a 
whole as well as its internal energy. Thus, for a steady flow of 
gas or liquid we have 

\Mv 2 + E+pV= constant 

or 

?Mv 2 + W = constant, 

where W and M are the enthalpy and the mass of a given quantity 
of substance and v the velocity of flow. The above equation 
signifies that the quantity \Mv 2 + W is the same for a given mass 
of substance no matter where it is in the flow. 

Where it may be necessary to take into account also the poten- 
tial energy in a field of gravity in the flow of a liquid (the weight 
is unimportant in gas flow), we can similarly write 

iMv 2 + Mgz +E+pV= constant, 

where z is the height of a given point in the flow. 

Let us assume that the motion in the flow is not accompanied by 
any appreciable friction, either within the flowing substance 



§61] STEADY FLOW 185 

itself or against any external obstacles; this is in a sense the 
opposite of a Joule-Kelvin process, where friction plays an 
important part. Under these conditions we can assume not only 
that the flow as a whole is thermally isolated from the external 
medium (as we have assumed throughout) but also that during 
the motion each individual element of substance is thermally 
isolated; if there were appreciable friction this would not be so, 
since frictional heat would be generated within the flow. That is, 
we may assume that during the motion each element of substance 
expands or contracts adiabatically. 

Let us consider, for example, the outflow of gas under these 
conditions from a vessel in which it is at a pressure p different 
from the atmospheric pressure p . If the outflow takes place 
through a sufficiently small opening, the velocity of the gas within 
the vessel may be taken as zero. The velocity v of the outflowing 
jet is given by the equation 

W+ 1 v 2 =Wq . 

here we have taken the mass M as 1 g, so that W and W are the 
enthalpies per gram of gas within the vessel and in the outflowing 
jet. If the gas is assumed ideal and its specific heat independent 
of temperature, then the formula c p = dWldT or dW = c p dT 
(cf. § 56) shows that W - W = c p ( T - T) , and hence 

v*=2c p (T -T). 

Finally, the temperature T in the outflowing jet can be expressed 
in terms of the temperature T of the gas in the vessel by means 
of the equation of adiabatic expansion of the gas derived in §59; 
this states that the product Tp-v-V' y is constant: 

T =T(pJpp-^y. 

Thus we finally obtain the following formula for the velocity of 
outflow of the gas: 

v 2 = 2c p T[(p l P yv-Viv-l]. 

The flow of liquids generally occurs without any appreciable 
change in their volume, owing to their comparatively small 



186 THERMAL PROCESSES [VIII 

compressibility. In other words, a flowing liquid may be regarded 
as incompressible and of constant density. 

The equation of steady (frictionless) flow of such a liquid is 
especially simple. In this case the general equation of an adiabatic 
process, dE+pdV = 0, reduces to dE = simply, since dV = 
owing to the incompressibility of the liquid. That is, the energy 
E remains constant and may therefore be omitted from the left- 
hand side of the equation 

\Mv 2 + E+pV+ Mgz = constant. 

Dividing this equation by the mass M and noting that the ratio 
M/V is the density p of the liquid, we finally deduce that the 
following quantity remains constant throughout a thermally 
isolated steady frictionless flow of an incompressible liquid: 

?v 2 + pip + gz = constant. 

This is called Bernoulli's equation. 

As an example, let us consider the motion of a liquid in a pipe 
of variable cross-section, which for simplicity we shall assume 
to lie horizontally. Then the force of gravity has no effect on the 
motion, and Bernoulli's equation gives 

\tf + p\p = iv 2 + polp, 

where v and v are the flow velocities at any two cross-sections 
of the tube, and p and p the corresponding pressures. If the 
areas of these two cross-sections are ,S and S, the volumes of 
liquid passing through them per unit time are v S and vS, and 
since the liquid is assumed incompressible v S = u 5 , or 

v = v S lS, 

i.e. the velocity of an incompressible liquid at any cross-section 
is inversely proportional to its area. Substituting this expression 
for v in Bernoulli's equation, we obtain a relation between the 
pressure and the cross-sectional area: 

P = P +2p(V 2 -V 2 ) 

= Po + %pv 2 (l-S 2 IS 2 ). 



§62] IRREVERSIBILITY OF THERMAL PROCESSES 187 

We see that the pressure is greater in the wider parts of the pipe 
than in the narrower ones. 

Let us now apply Bernoulli's equation to determine the velocity 
of a jet of liquid leaving a vessel through a small opening. Since 
the area of the opening is assumed small in comparison with the 
cross-section of the vessel, we may neglect the fall in the level 
of the liquid in the vessel. Using also the fact that the pressure 
on the surface of the liquid in the vessel and the pressure in the 
jet are the same, and equal to the atmospheric pressure, we 
obtain from Bernoulli's equation 

faP + gZi = gz 2 , 

where v is the velocity of the outflowing jet, and z 2 and z x the 
heights of the surface of the liquid in the vessel and the point of 
outflow of the liquid; hence 

v = y/{2gh), 

where h = z 2 — Z\- This formula, called Torricelli's formula, 
shows that the velocity of the outflowing liquid from a small 
aperture is the same as the velocity of fall of a body from a 
height h which is equal to the height of the liquid in the vessel 
above the aperture. 

§62. Irreversibility of thermal processes 

The mechanical movements of material bodies, occurring in 
accordance with the laws of mechanics, have the following 
remarkable property. Whatever the motion of a body, the reverse 
motion is always possible, i.e. the motion in which the body passes 
through the same points in space with the same velocities as in 
the original motion, but in the opposite direction. For example, 
let a body be projected in the field of gravity at a certain angle to 
the horizontal; it will describe a certain trajectory and fall to the 
ground at some point. If now the body is projected from this 
point at the angle at which it fell and at the corresponding velocity, 
it will describe the same trajectory in the opposite direction and 
fall to its original position (if air friction may be neglected). 

This reversibility of mechanical motions may be alternatively 
formulated by saying that they are symmetrical as regards inter- 



188 THERMAL PROCESSES [VIII 

changing the future and the past, i.e. with respect to time reversal. 
The symmetry of mechanical motions follows at once from the 
equations of motion themselves, since when the sign of the time 
is reversed so is that of the velocity, but the acceleration is left 
unchanged. 

The situation is quite different as regards thermal phenomena. 
If a thermal process takes place, then the reverse process (i.e. the 
process in which the same thermal states are traversed in the 
opposite order) is in general impossible. Thus thermal processes 
are as a rule irreversible. 

For example, if two bodies at different temperatures are 
brought into contact, the hotter body will transmit heat to the 
colder body, but the reverse process (a spontaneous direct 
transfer of heat from the colder to the hotter body) never occurs. 

The expansion of a gas into a vacuum, described in §59, is 
likewise an irreversible process. The gas spreads through the 
opening on both sides of the partition, but without external 
interference it will never collect spontaneously in one half of the 
vessel again. 

Any system of bodies left to itself tends to reach a state of 
thermal equilibrium, in which the bodies are at relative rest, with 
equal temperatures and pressures. Having reached such a state, 
the system will not of its own accord leave that state. In other 
words, all thermal phenomena accompanied by processes of 
approach to thermal equilibrium are irreversible. 

For instance, all processes accompanied by friction between 
moving bodies are irreversible. The friction causes a gradual 
slowing down of the motion (the kinetic energy being converted 
into heat), i.e. an approach to a state of equilibrium in which 
there is no motion. For this reason, in particular, a Joule-Kelvin 
process, in which the gas passes through an obstacle with a large 
amount of friction, is irreversible. 

All thermal processes occurring in Nature are to some extent 
irreversible. In some cases, however, the degree of irreversibility 
may be so slight that the process may be regarded as reversible 
with sufficient accuracy. 

It is clear from the foregoing that, in order to achieve rever- 
sibility, it is necessary to eliminate from the system as far as 
possible all processes which constitute an approach to thermal 
equilibrium. For example, there must be no direct transfer of 



§62] IRREVERSIBILITY OF THERMAL PROCESSES 189 

heat from a hotter to a colder body and no friction in the motion 
of bodies. 

An example of a process which is reversible to a high degree 
(and, in the ideal case, perfectly reversible) is the adiabatic ex- 
pansion or compression of a gas described in §59. The condition 
of thermal isolation excludes a direct exchange of heat with the 
surrounding medium. The "sufficiently slow" movement of the 
piston ensures that there are no irreversible processes of expan- 
sion of a gas into the vacuum which would be produced behind 
a too rapidly moving piston, since this is what the condition of 
slowness signifies. Of course, in practice there will still remain 
some causes of irreversibility (imperfect thermal isolation of the 
vessel containing the gas; friction in the movement of the piston). 

"Slowness" is a general characteristic of reversible processes: 
the process must be so slow that the bodies involved in it are 
able to reach at every instant the state of equilibrium which 
corresponds to the prevailing external conditions. In the example 
of the expansion of a gas, the latter must be able to follow the 
piston and remain homogeneous throughout its volume. Com- 
plete reversibility could be achieved only in the ideal case of an 
infinitely slow process, and for this reason alone a process 
occurring at a finite rate cannot be completely reversible. 

We have already mentioned that, in a system of bodies in 
thermal equilibrium, no process can take place without external 
interference. This can be stated in another way: bodies in thermal 
equilibrium can do no work, since work requires mechanical 
motion, i.e. a conversion of energy into the kinetic energy of the 
bodies. 

This extremely important assertion that work can not be 
obtained from the energy of bodies in thermal equilibrium is 
called the second law of thermodynamics. We are always sur- 
rounded by considerable sources of thermal energy in a state 
close to equilibrium. An engine working merely on the energy 
of bodies in thermal equilibrium would constitute a perpetual- 
motion machine. The second law of thermodynamics prevents the 
construction of such a perpetual-motion machine of the second 
kind, just as the first law of thermodynamics (the law of conserva- 
tion of energy) prevents that of a machine of the first kind, i.e. one 
which would do work "from nothing", without any external 
source of energy. 



190 



THERMAL PROCESSES 



[VIII 



§63. The Carnot cycle 

From the foregoing it follows that work can be done only by 
means of a system of bodies which are not in thermal equilibrium 
with one another. Let us imagine such a system idealised as two 
bodies at different temperatures. If the two bodies are simply 
brought into contact, heat will pass from the hotter to the colder 
body, but no work will be done. The transfer of heat from a hotter 
to a colder body is an irreversible process, and this example 
demonstrates the general rule that irreversible processes prevent 
the doing of work. 

If it is desired to obtain the maximum possible work from 
given bodies, the process must be made as nearly reversible as 
possible: all irreversible processes must be avoided, and only 
processes which occur to the same extent in both directions must 
be used. 




Fig. 80. 



Returning to the system of the two bodies, we denote their 
temperatures by T 1 and T 2 (and let T 2 > T t ), and conventionally 
call the hotter body a heat source, and the colder body a heat 
sink. Since direct exchange of heat between these bodies is not 
permissible, it is clear first of all that, in order to do work, a 
further body must be used; this will be called the working medium. 
It may be imagined as a cylindrical vessel containing gas and 
closed by a piston. 

We shall represent the process in which the medium takes 
part, using a pV diagram (Fig. 80). Let the gas be initially at a 
temperature T 2 , and let its state be represented by the point A 
in the diagram. The working medium is now brought into contact 
with the heat source, and the gas is caused to expand; it gains a 



§63] THE CARNOT CYCLE 191 

certain quantity of heat from the heat source, while remaining 
at the source temperature T 2 (the total quantity of heat in the 
heat source is assumed so large that its temperature is not 
changed when a small quantity of heat is transferred to the gas). 
Thus the gas undergoes a reversible isothermal expansion, since 
heat is transferred only between bodies at the same temperature. 
In Fig. 80 this process is shown by the isotherm AB. 

Next, the working medium is removed from the heater, ther- 
mally isolated and further expanded, this time adiabatically. 
In this expansion the gas is cooled, and the expansion is continued 
until the temperature of the gas falls to the temperature T x of 
the heat sink. This process is represented in the diagram by the 
adiabatic BC, which is steeper than the isotherm AB, since in 
adiabatic expansion the pressure falls more rapidly than in 
isothermal expansion. 

The working medium is now brought into contact with the 
heat sink and the gas is isothermally compressed at temperature 
T lt thereby transferring a certain quantity of heat to the heat 
sink. Finally, the working medium is removed from the heat 
sink and the gas is adiabatically compressed to return it to its 
initial state; for this purpose it is necessary to make the proper 
choice of the point D, i.e. the volume to which the isothermal 
compression CD is taken. 

Thus the working medium undergoes a cyclic process, return- 
ing to its original state but doing a certain quantity of work 
represented by the area of the curvilinear quadrilateral ABCD. 
This work is done by virtue of the fact that on the upper isotherm 
the working medium takes from the heat source a greater quantity 
of heat than it gives to the heat sink on the lower isotherm. 
Every stage of this cyclic process is reversible, and the work 
done is therefore the maximum possible for a given quantity of 
heat taken from the source. 

The process just described is called a Carnot cycle. It shows 
that, in principle, work can be done reversibly by means of two 
bodies at different temperatures. Being the maximum possible 
amount, this work is independent of the properties of the working 
medium. 

The ratio of the work done to the quantity of energy taken 
from the hotter body is called the efficiency of the heat engine 
and will be denoted by 17. It is clear from the above that the 



192 THERMAL PROCESSES [VIII 

efficiency of a Carnot cycle is the maximum possible for any heat 
engine operating with given temperatures of the heat source and 
sink. It can be shown (see §65) that this efficiency is 

17max = (^2-7Y)/^2- 

Thus, even in the ideal limit of completely reversible operation 
of a heat engine, the efficiency is less than unity: a fraction TJT 2 
of the energy taken from the heat source is transferred un- 
profitably to the heat sink as heat. This fraction decreases with 
increasing temperature T 2 for given 7\. The temperature 7\ 
is usually that of the surrounding air, and therefore cannot be 
reduced. To decrease the fraction of energy wasted, therefore, 
the aim in applications is to operate the engine at the maximum 
possible temperature T 2 . 

The efficiency of an actual heat engine is always less than 
Tj max because of the irreversible processes which unavoidably 
occur in it. The quantity Tj/T/ max , i.e. the ratio of the efficiency of 
the actual engine to that of an ideal engine with the same heat 
source and sink temperatures, can be used to represent the degree 
to which the engine approaches the ideal one. This is therefore 
the ratio of the work done by the heat engine to the maximum 
work which could be obtained in the given conditions if the engine 
were operating reversibly. 

§64. The nature of irreversibility 

All thermal phenomena reduce ultimately to the mechanical 
movement of the atoms and molecules in a body. The irrever- 
sibility of thermal processes is therefore, at first sight, in conflict 
with the reversibility of all mechanical motions. This contradiction 
is in fact only apparent. 

Suppose that a body slides on another body. Because of 
friction, this motion will be gradually slowed down and the 
system will finally reach a state of thermal equilibrium; the 
motion will then cease. The kinetic energy of the moving body 
is converted into heat in this process, i.e. kinetic energy of the 
random motion of the molecules in both bodies. This conversion 
of energy into heat can obviously be brought about in an infinite 
number of ways: the kinetic energy of the motion of the body 
as a whole can be distributed between the enormous number of 



§64] THE NATURE OF IRREVERSIBILITY 193 

molecules in an enormous number of ways. In other words, the 
state of equilibrium in which there is no macroscopic motion can 
occur in an immensely greater number of ways than a state in 
which a considerable quantity of energy is concentrated in the 
form of kinetic energy of the ordered motion of the body as a 
whole. 

Thus the change from a non-equilibrium state to an equilibrium 
state is a change from a state which can occur in a small number 
of ways to one which can occur in a very much larger number of 
ways. It is clear that the most probable state of a body (or system 
of bodies) is that which can occur in the largest number of ways, 
and this will be the state of thermal equilibrium. Thus, if a system 
left to itself (i.e. a closed system) is not in a state of equilibrium, 
then its subsequent behaviour will almost certainly be to enter a 
state which can occur in a very large number of ways, i.e. to 
approach equilibrium. 

On the other hand, when a closed system has reached a state 
of equilibrium, it is most unlikely to leave that state spontaneously. 

Thus the irreversibility of thermal processes is probabilistic. 
The spontaneous passage of a body from an equilibrium state to 
a non-equilibrium state is, strictly speaking, not impossible, but 
only very much less probable than that from a non-equilibrium 
state to an equilibrium state. The irreversibility of thermal pro- 
cesses is ultimately due to the very large number of molecules of 
which bodies are composed. 

The improbability of a body's spontaneously leaving an equilib- 
rium state may be judged by considering the expansion of a gas 
into a vacuum. Let the gas be initially in one half of a vessel 
divided by a partition into two equal parts. When an opening is 
made in the partition, the gas spreads uniformly through both 
parts of the vessel. The opposite transfer of the gas into one half 
of the vessel will never occur without external interference. The 
reason for this is easily seen by a simple calculation. Each 
molecule of gas, in its motion, spends on average the same time 
in each part of the vessel; we may say that the probability of 
finding it in either half of the vessel is i If the gas may be regarded 
as ideal, its molecules move independently. The probability of 
finding two given molecules in the same half of the vessel at the 
same time is therefore i . £ = i; the probability of finding all 
N molecules of gas in one half of the vessel is 2~ N . For instance, 



194 THERMAL PROCESSES [VIII 

with a relatively small quantity of gas, containing say 10 20 
molecules, this probability is given by the fantastically small 
number 2 -1020 ~ io _3xl ° 19 . In other words, this occurrence would 
be observed about once in a time represented by the number 
10 3X1 ° 19 — whether seconds or years is immaterial, since a second, 
a year, and indeed the time the Earth has existed, are equally 
small in comparison with this vast interval of time. 

A similarly small number (10 _3X1 ° 10 ) may be shown to represent 
the probability that a single erg of heat will pass from a body 
at 0°C to another body at 1°C. 

It is clear from these examples that the possibility of any 
appreciable spontaneous reversal of a thermal process is in 
essence a pure abstraction: its probability is so small that the 
irreversibility of thermal processes may in practice be regarded 
as exactly true. 

The probabilistic nature of irreversibility appears, however, 
in the fact that in Nature there are nevertheless spontaneous 
deviations from equilibrium, although these are very small and 
short-lived; they are called fluctuations. Owing to fluctuations, 
for example, the density and temperature in different small 
regions of a body in equilibrium are not exactly constant, but 
undergo some very slight variations. For instance, the temper- 
ature of 1 milligram of water in equilibrium at room temperature 
will vary by amounts of the order of 10 -8 degree. There are also 
phenomena in which fluctuations play an important part. 

§65. Entropy 

A quantitative characteristic of the thermal state of a body, 
which describes the degree to which it tends to enter other 
states, is the number of microscopic ways in which the state 
can occur. This number is called the statistical weight of the 
state and will be denoted by T. A body left to itself will tend 
to enter a state of greater statistical weight. 

It is customary, however, to use instead of the number T 
itself its logarithm multiplied by Boltzmann's constant k. The 
quantity thus defined, 

S = k log e r, 
is called the entropy of the body. 



§65] ENTROPY 195 

The number T of ways in which a state of a system consisting 
of, for example, two bodies can occur is evidently equal to the 
product of the numbers T t and T 2 of ways in which the state of 
each body separately can occur: T = I^IY Hence 

S = k log e r 
= k log e T x + k logg T 2 

= s t +s 2 . 

Thus the entropy of a composite system is equal to the sum of 
the entropies of its parts, and it is for this reason that the logarithm 
is used in the definition of the entropy. 

The law which governs the direction of thermal processes may 
be formulated as a law of increase of entropy: in all thermal 
processes occurring in a closed system, the entropy of the 
system increases, and the maximum possible value of the 
entropy of a closed system is reached in a state of thermal 
equilibrium. This is a more precise quantitative form of the 
second law of thermodynamics. The law was stated by Clausius, 
and its interpretation in terms of molecular kinetics was given 
by Boltzmann. 

Conversely, we may say that any process in which the entropy 
of a closed system increases is irreversible; the greater the 
increase in entropy, the higher the degree of irreversibility. 
The ideal case of a completely reversible process corresponds 
to that in which the entropy of a closed system remains 
constant. 

A precise definition of what is meant by the "number of 
microscopic ways" in which a thermal state of a body can occur 
is given in statistical physics, and only when this has been done 
is it possible to carry out an actual calculation of the entropy 
of various bodies and to establish the relation between it and 
other thermal quantities. 

A more detailed theoretical analysis makes it possible to 
derive a relation which is fundamental in thermodynamic 
applications of the concept of entropy. This relation is one 
between the change dS in the entropy of a body in an infinitesimal 
reversible change in state and the quantity of heat dQ which it 
gains in the process; the body is, of course, assumed not closed, 



196 THERMAL PROCESSES [VIII 

so that the reversibility of the process does not require that its 
entropy should be constant. The relation is 

dS = dQ/T, 

where T is the temperature of the body. 

The existence of a relation between dS and dQ is entirely 
reasonable. When the body gains heat, the thermal motion of its 
atoms is increased, i.e. their distribution over various states of 
microscopic motion becomes more random, and so the statistical 
weight increases. It is also reasonable that the effect of a given 
quantity of heat on the thermal state of the body is described 
by the relative magnitude of this quantity of heat and the total 
internal energy of the body, and hence decreases with increasing 
temperature. 

The relation dQ — T dS leads, in particular, to the expression 
already given in §63 for the efficiency of a Carnot cycle. We have 
seen that this process involves three bodies: a heat source, a 
heat sink and a working medium. The latter is returned to its 
initial state by the cycle, and its entropy therefore also returns 
to its original value. The condition for the process to be rever- 
sible, i.e. the requirement that the total entropy of the system 
should be unchanged, therefore demands that the sum of the 
entropies 5 X of the heat sink and S 2 of the heat source should 
be constant. Let the sink gain a small quantity of heat A(2i in 
the cycle, and let the source lose AQ 2 . Then 

AS X + AS 2 = A^m- A<2 2 /T 2 = 0, 

whence A(2i = T 1 AQ 2 /T 2 . The work done in one cycle is A = 
AQ 2 — A(?i, and the efficiency is therefore 

ti=A^Q 2 = \-TJT 2 . 



CHAPTER IX 



PHASE TRANSITIONS 



§66. Phases of matter 

The evaporation of a liquid and the melting of a solid are pro- 
cesses of the type which are called in physics phase transitions. 
The characteristic feature of these processes is that they are 
discontinuous. For example, when ice is heated, its thermal 
state changes gradually until the temperature 0°C is reached 
and the ice suddenly begins to change into liquid water, which 
has entirely different properties. 

States of matter between which phase transitions occur are 
called phases of matter. In this sense the aggregate states of 
matter (gaseous, liquid and solid) are different phases. For 
example, ice, liquid water and steam are the phases of water. 
The concept of phases, however, is broader than that of aggregate 
states; we shall see that different phases can exist within a single 
aggregate state. 

It must be emphasised that, in speaking of the solid state as a 
separate phase of matter (distinct from the liquid phase), we are 
considering only the crystalline solid state. An amorphous solid 
is transformed on heating into a liquid by a gradual softening 
without discontinuity, as already described in §52; the amorphous 
solid state is therefore not a separate phase of matter. For 
instance, solid and liquid glass are not distinct phases. 

The transition from one phase to another always occurs at a 
fixed temperature (at a given pressure). For example, ice begins 
to melt at 0°C (at atmospheric pressure) and on further heating 
the temperature remains constant until all the ice is changed 
into water. During this process, ice and water coexist in contact. 

This exhibits another aspect of the temperature of a phase 
transition: it is the temperature at which there is thermal equilib- 
rium between the two phases. In the absence of external inter- 
actions (including an external heat supply) the two phases 

197 



198 PHASE TRANSITIONS [iX 

can coexist indefinitely at this temperature. At temperatures 
above or below the transition point, however, only one or the 
other phase can exist. For instance, at a temperature below 
0°C only ice can exist (at atmospheric pressure), and above 
0°C only liquid water. 

When the pressure changes, so does the phase-transition 
temperature. In other words, a phase transition occurs when the 
pressure and temperature of the substance satisfy a certain 
fixed relation. This relation may be represented graphically as 
a curve in what is called a phase diagram, whose coordinates 
are the pressure p and the temperature T. 

Liquid 

Gas 



Fig. 81. 

Let us consider, as an example, a phase transition between a 
liquid and its vapour. The phase-transition curve (called in this 
case the evaporation curve) determines the conditions under 
which the liquid and the vapour can coexist in equilibrium. The 
curve divides the plane into two parts, one of which corresponds 
to states of one phase and the other to states of the other phase 
(Fig. 81). Since in this case, at a given pressure, the higher 
temperatures correspond to the vapour and the lower ones to 
the liquid, the region to the right of the curve corresponds to 
the gaseous phase, and the region to the left corresponds to the 
liquid phase. The points on the curve itself correspond, as 
already mentioned, to states in which two phases coexist. 

The phase diagram can be drawn not only in the pT plane but 
also in other coordinates: p and V, or T and V, where V is the 
volume of a given quantity of matter. We shall take V to be the 
specific volume, i.e. the volume of unit mass of matter (so that 
l/V is the density of the substance). 

Let us consider the phase diagram in the VT plane, and a gas 
whose specific volume and temperature correspond to some 
point a in Fig. 82. If the gas is compressed at constant tempera- 



§66] 



PHASES OF MATTER 



199 



ture, then the point representing the state of the gas will move to 
the left along a straight line parallel to the V axis. At a certain 
pressure corresponding to the specific volume V g (the point A), 
the gas begins to condense into a liquid. As the system is com- 
pressed further, the quantity of liquid increases and the quantity 
of gas decreases; finally, when a certain point B is reached the 
substance is entirely liquid and its specific volume is V t . 




The specific volumes of the gas and the liquid which are formed 
from each other (V g and V t ) are functions of the temperature at 
which the transition occurs. When these two functions are 
represented by appropriate curves, we obtain a phase diagram of 
the kind shown in Fig. 82. The regions of the diagram to the 
right and left of the hatched area correspond to the gaseous and 
liquid phases. The hatched area between the two curves is the 
region of separation into two phases. The horizontal hatching 
is significant: the points A and B at which a horizontal line 
through a point C in this region meets the boundaries of the region 
give the specific volumes of the liquid and vapour coexisting at 
that point. 

The different points on AB evidently correspond to equilibrium 
of the same liquid and vapour in different relative amounts. Let 
the fractions of vapour and liquid at some point C be x and 1 — x. 
Then the total volume of the system per unit mass is 



V = xV a +(l-x)V h 



whence 



_ V-Vi t _ 
x yr l x 



v g -v; 



200 PHASE TRANSITIONS [IX 

The ratio of these quantities is 

x ^ V-V l = BC 

\-x V g -V AC " 

We see that the quantities of vapour and liquid are inversely 
proportional to the lengths of AC and BC, i.e. the distances of 
C from the points A and B which correspond to the pure vapour 
and pure liquid. This relation is called the lever rule. 

The phase diagram with pressure instead of temperature as 
ordinate is exactly similar in appearance. We see that these dia- 
grams do not resemble the diagrams in the pT plane. The region 
of separation into two phases, which in the pT diagram is only a 
line, occupies a whole area in the VT and Vp diagrams. This 
difference arises because phases in equilibrium necessarily have 
the same temperature and pressure by the general conditions of 
thermal equilibrium, but their specific volumes are different. 

Table 1 shows the melting and boiling points of a number of 
substances (at atmospheric pressure). 

Table 1 





Melting point 


Boiling point 




(°C) 


(°C) 


Helium-3 


_ 


-270-0 (3 -2°K) 


Helium-4 


- 


-268-9 (4-2°K) 


Hydrogen 


-259-2 (14°K) 


-252-8 (20-4°K) 


Oxygen 


-219 


-183 


Ethyl alcohol 


-117 


78-5 


Ethyl ether 


-116 


34-5 


Mercury 


- 38-9 


356-6 


Lead 


327 


1750 


Aluminium 


660 


2330 


Sodium chloride 


804 


1413 


Silver 


961 


2193 


Copper 


1083 


2582 


Iron 


1535 


2800 


Quartz 


1728 


2230 


Platinum 


1769 


4000 


Tungsten 


3380 


6000 



Helium liquefies at a lower temperature than any other sub- 
stance existing in Nature; the solidification of helium will be 
discussed in §72. Tungsten has higher melting and boiling points 
than those of any other chemical element. 



§67] 



THE CLAUSIUS-CLAPEYRON EQUATION 



201 



§67. The Clausius-Clapeyron equation 

The transition of matter from one phase to another always 
involves the gain or loss of a certain quantity of heat called the 
latent heat or heat of transition. When a liquid becomes a gas 
this is the heat of evaporation; when a solid becomes a liquid, it is 
the heat of fusion. 

Since a phase transition occurs at constant pressure, the heat 
of transition q 12 from phase 1 to phase 2 is equal to the difference 
of the enthalpies W x and W 2 of the substance in the two phases 
(see §56): 

Qt2=W 2 -W l . 

It is clear that q 12 = — q 21 , i.e. if heat is absorbed in a given phase 
transition, the reverse transition is accompanied by the evolution 
of heat. 

In melting and in evaporation, heat is absorbed. These are 
particular cases of a general rule according to which a phase 
transition brought about by heating is always accompanied by the 
absorption of heat. This rule in turn is a consequence of Le 
Chatelier's principle: heating tends to cause processes to occur 
which are accompanied by absorption of heat and which therefore 
as it were act against the external interaction. 



i p 





(a) 



(b) 



Fig. 83. 



The same principle can be used to relate the direction of the 
phase-equilibrium curve in the pT plane to the change in volume 
in a phase transition. Let us consider, for example, an equilibrium 
system consisting of a liquid and a vapour and suppose that it is 
compressed, so that the pressure in it increases. Then processes 
must occur in the system which reduce the volume of the sub- 
stance and thus counteract the effect of the compression. For 



202 



PHASE TRANSITIONS 



[IX 



this to be so, condensation of the vapour must occur, since the 
conversion of vapour into liquid is always accompanied by a 
decrease in volume. This means that, as we move upwards from 
the equilibrium curve (Fig. 83), we must enter the region of the 
liquid phase. In this case the liquid is also the "low-temperature" 
phase, i.e. the phase which exists at lower temperatures. Thus it 
follows that the equilibrium curve for a liquid and a gas must 
have the form shown in Fig. 83a, and not that in Fig. 83b; the 
transition temperature must increase with increasing pressure. 

The same relation between transition temperature and pressure 
must evidently occur whenever the transition to the "high- 
temperature" phase is accompanied by an increase in volume. 
For example, since in almost all cases the volume of a substance 
increases on melting, the melting point usually rises with increas- 
ing pressure. In some substances, however, melting is accom- 
panied by a decrease in volume (as for instance in ice, cast iron, 
and bismuth). For these substances the melting point is lowered 
by increasing the pressure. 

All these qualitative results are expressed quantitatively by a 
formula which relates the slope of the phase-equilibrium curve, 
the heat of transition, and the change in volume in the transition. 



p i 



,/fc^ 



Fig. 84. 



To derive this formula, let us imagine a very "narrow" Carnot 
cycle applied to a certain quantity of substance, the isothermal 
processes being a transition of the substance from phase 2 to 
phase 1 at a pressure p and the reverse transition from phase 1 
to phase 2 at a pressure p + dp. These transitions are represented 
in the pV phase diagram (Fig. 84) by the lines ab and cd. The 
sides be and da should, strictly speaking, be taken as segments of 
adiabatics, but in the limit of an infinitely narrow cycle the 



§68] EVAPORATION 203 

difference is unimportant and does not affect the area of the cycle, 
which is the work done in the cyclic process; this work is evidently 
just (V 2 — Vi) dp. The work must also be equal to the product of 
the quantity of heat q 12 which is expended (on the isotherm cd) 
and the efficiency of the Carnot cycle. The quantity q 12 is simply 
the heat of transition from phase 1 to phase 2, and the efficiency 
is dT/T, where dT is the temperature difference between the two 
isotherms. Thus we have 

(V 2 -V 1 )dp = q l2 dTlT 
or 

dp _ <?i2 
dT T(V 2 -VS 

This formula, which determines the slope of the phase- 
equilibrium curve p = p(T), is called the Clausius-Clapeyron 
equation. It may also be written in the form 

dT_ T(V 2 -V 1 ) 
dp q 12 

where the temperature of the transition is regarded as a function 
of pressure. In these formulae the volumes V x , V 2 of the two 
phases and the heat q X2 relate to a given quantity of the substance 
(e.g. one gram or one gram-molecule). 

It should be noted that the derivative dp/dT is inversely pro- 
portional to the difference in volume V 2 — V x . Since the change 
in volume in evaporation is large and that in melting is small, 
melting curves are much steeper than evaporation curves. For 
example, to lower the boiling point of water by 1° it is sufficient 
to reduce the pressure by 27 mm Hg, whereas the same change in 
the melting point of ice would require the pressure to be increased 
by 130atm. 

§68. Evaporation 

A vapour in equilibrium with its liquid is said to be saturated, 
and its pressure is called the saturated vapour pressure. The 
liquid-vapour equilibrium curve (Fig. 81, §66) may also be 



204 PHASE TRANSITIONS [IX 

regarded as showing the relation between this pressure and the 
temperature. 

The saturated vapour pressure always increases with increas- 
ing temperature. We have seen above that this behaviour is due 
to the increase in volume of a substance on evaporation. This 
increase is usually very large. For example, the volume of water 
vapour at 100°C is 1600 times the volume of water; the boiling 
of liquid oxygen at — 183°C is accompanied by a volume increase 
by a factor of about 300. 

At sufficiently low temperatures the density of the saturated 
vapour becomes so small that it behaves as an ideal gas. A simple 
formula can then be derived for the temperature dependence of 
the vapour pressure. To do so, we use the Clausius-Clapeyron 
equation, 

dp q 



dT T(V a -V l y 

with q the molar heat of evaporation, and V g and V x the molar 
volumes of the vapour and the liquid. Since the volume V g is 
very large in comparison with V t , the latter may be neglected. 
The volume of one gram-molecule of gas is V g = RT/p. We have 

dp pq 



dT RT 2 ' 



or 



)_dp__ d \o% e P _ q 
pdT~ dT ~ RT 2 ' 

Although the heat of evaporation is itself a function of tem- 
perature, it may often be regarded as practically constant over 
considerable ranges of temperature; for example, the heat of 
evaporation of water decreases by only 1 0% between and 1 00°C. 
The above formula may then be written as 

d\og e P = d_ (_q 

dT dT\RT 

whence 

loge/? = constant — qlRT 



§68] EVAPORATION 205 

and finally 

p = ce -«RT^ 

where c is a constant coefficient. According to this formula the 
saturated vapour pressure increases very rapidly (exponentially) 
with temperature. 

The origin of this exponential dependence may be understood 
as follows. The molecules in a liquid are held together by cohesion 
forces; to overcome these forces and transfer a given molecule 
from the liquid to the vapour, work must be done. We may say 
that the potential energy of a molecule in the liquid is less than 
its potential energy in the vapour by an amount equal to the heat 
of evaporation per molecule. If q is the molar heat of evaporation, 
this difference of potential energies is qlN , where N is 
Avogadro's number. 

We can now use Boltzmann's formula (§54) to show that the 
increase in the potential energy of a molecule by q/N decreases 
the gas density by a factor e~ qm ° kT = e~ qlRT in comparison with 
the liquid. The pressure of the vapour is proportional to this 
expression. 

The following are the values of the heats of evaporation and 
heats of fusion for various substances at atmospheric pressure, 
in joules per mole: 





<7ev 


<?fu 


Helium 


80 





Water 


40 500 


5 980 


Oxygen 


6 800 


442 


Ethyl alcohol 


39 000 


4 800 


Ethyl ether 


59 000 


7 500 


Mercury 


28 000 


2 350 



[It may be noted that from the heat of evaporation (far from the 
critical point; see §69) we can estimate the magnitude of the van 
der Waals forces between the molecules. As has been mentioned 
in §39, it is these forces which bring about the condensation of 
a substance. Thus, on dividing q ev by Avogadro's number to 
obtain the heat of evaporation per molecule, we derive a quantity 
which is a measure of the depth of the minimum on the curve of 



206 PHASE TRANSITIONS [IX 

van der Waals interaction. The quantity obtained in this way for 
helium is about one-hundredth of an electron-volt, and for the 
other liquids in the table it is between one-tenth and a few tenths 
of an electron- volt.] 

In ordinary conditions there is present over the surface of a 
liquid not only its own vapour but also another gas, namely air. 
This has little effect on the phase equilibrium; evaporation 
continues until the partial pressure of the vapour becomes equal 
to the saturated vapour pressure at the temperature of the liquid. 

The presence of the atmosphere considerably affects the 
evaporation process, however, which has a completely different 
form according as the saturated vapour pressure at a given tem- 
perature is less than or greater than the total pressure on the 
liquid. 

In the former case, the liquid evaporates comparatively slowly 
from its surface. It is true that the partial pressure of the vapour 
just above the surface almost immediately becomes equal to the 
saturated vapour pressure, but this saturated vapour penetrates 
into the surrounding space only slowly (by diffusion), and further 
liquid evaporates only as this vapour mixes with the air. The rate 
of evaporation is, of course, increased by artificially removing 
the vapour from the surface of the liquid. 

The process is different when the saturated vapour pressure 
becomes equal to or slightly greater than the ambient pressure 
and the liquid boils violently. This is shown by the intensive 
formation, on the surface of the vessel, of gas bubbles which 
grow by the evaporation of liquid into the bubbles and then 
become detached and rise through the liquid, causing mixing 
of it; a stream of vapour passes from the free surface of the liquid 
into the surrounding medium. 

For reasons which will be discussed later (see §99), the con- 
version of a liquid into a vapour cannot in general occur by 
spontaneous generation of vapour bubbles within a pure liquid. 
The centres of formation of the gas phase are tiny bubbles of 
other gases which already exist on the vessel walls or are formed 
thereon (or on particles suspended in the liquid) from gases 
dissolved in the liquid which are expelled on heating. Until the 
boiling point is reached (at which the saturated vapour pressure 
becomes equal to the external pressure), the pressure of the 
surrounding liquid prevents the growth of these bubbles. 



§69] THE CRITICAL POINT 207 

By careful previous purification and degassing of the liquid and 
the vessel walls it is possible to eliminate practically all vaporisa- 
tion centres in it (as may also happen during the boiling process 
itself). This leads to superheating, the liquid remaining a liquid at 
temperatures above the boiling point. On the other hand, in order 
to avoid superheating and to ensure that boiling occurs, various 
artificial sources of vaporisation centres are placed in the vessel 
of liquid (porous objects, pieces of glass capillary, and so on). 

A superheated liquid (i.e. a liquid at a temperature at which it 
would be expected to have become a gas, at the pressure con- 
cerned) is an example of what are called metastable states. 
These are states of limited stability. Although they can exist 
(when suitable precautions are taken) for a longer or shorter 
time, the equilibrium is relatively easily destroyed and the sub- 
stance enters a different state, which is stable. For example, a 
superheated liquid boils instantaneously when vaporisation 
centres are created in it. 

Similar phenomena occur in the reverse process of condensa- 
tion of a vapour. Here again the occurrence of the phase transi- 
tion, in the absence of liquid in contact with the vapour, requires 
the existence of condensation centres in the vapour, usually in 
the form of small impurities, as will be further discussed in §99. 
For this reason supercooling or supersaturation of a vapour is 
possible, in which it is brought into a state where the pressure 
exceeds the saturated vapour pressure at the temperature con- 
cerned. Such states can be reached, for example, by cooling a 
carefully purified saturated vapour by adiabatic expansion. 

§69. The critical point 

As the temperature rises, the saturated vapour pressure in- 
creases rapidly, and so does the density of the vapour, approach- 
ing that of the liquid. At a certain temperature the density of the 
vapour becomes equal to that of the liquid, and the vapour and 
liquid become indistinguishable. In other words, the equilibrium 
curve of the liquid and gas in the pT phase diagram terminates at 
some point (K in Fig. 85). This is called the critical point, and its 
coordinates are called the critical temperature T c and critical 
pressure p c of the substance. 

In the VT diagram (and similarly in the Vp diagram) the 
approach to the critical point is shown by the approach to equality 



208 



PHASE TRANSITIONS 



[IX 



of the specific volumes of the liquid and vapour as the tempera- 
ture increases, i.e. by the approach of the two curves which form 
the boundaries of the hatched region in Fig. 82 (§66). For T = T c 
the two curves join, and we thus have essentially a single smooth 
curve with a maximum at K (Fig. 86). This is the critical point, 
its coordinates being the critical temperature T c and the critical 
specific volume V c . 

As the properties of the liquid approach those of the gas, the 
heat of transition q between them decreases, and becomes zero 
at the critical point. 





Fig. 85. 



Fig. 86. 



The existence of the critical point very clearly demonstrates 
that there is no fundamental difference between the liquid and 
gaseous states of matter. For, when considering any two states 
a and b (Fig. 86) of very different density, we call the denser 
state b the liquid state, and the less dense a the gaseous state. By 
compressing the gas a at constant temperature we can convert it 
to the liquid b, passing through a stage of separation into two 
phases. But the passage between the same states a and b can also 
be carried out by first raising and then lowering the temperature 
while the volume is decreased, in such a way as to move along 
a path in the VT plane which passes above the critical point, as 
shown by the broken line in Fig. 86. In this case there is no 
discontinuous change of state anywhere, the substance remains 
homogeneous, and we can not say that the substance ceases at 
some point to be a gas and becomes a liquid. 

From the diagram in the VT plane we can easily find what 
happens on heating a closed vessel (for example, a sealed tube) 



§69] THE CRITICAL POINT 209 

containing a certain quantity of liquid and the vapour above it. 
Since the total volume of the substance is constant, this will 
correspond to movement upward along a vertical line in the VT 
plane. If the volume of the tube exceeds the critical volume 
corresponding to the given quantity of substance, this line will 
lie to the right of the critical point (AB in Fig. 86) and as heating 
proceeds the quantity of liquid will decrease until the whole of 
the substance is converted into vapour (at B); the boundary 
(meniscus) between the liquid and the vapour will disappear at 
the lower end of the tube. If the volume of the tube is less than 
the critical volume (point A'), vapour will condense on heating 
until the whole of the substance becomes liquid (at B')\ the 
meniscus will disappear at the top of the tube. Finally, if the 
volume of the tube is equal to the critical volume, the meniscus 
will disappear somewhere within the tube at the critical 
temperature T c . 

The values of the absolute critical temperatures T c , pressures 
p c and densities p c for a number of substances are shown in 
Table 2. 





Table 2 








T C (°K) 


Pc (atm) 


p c (g/cm 3 ) 


Water 


647-2 


218-5 


0-324 


Alcohol 


516-6 


63-1 


0-28 


Ether 


467-0 


35-5 


0-26 


Carbon dioxide 


304-2 


73-0 


0-46 


Oxygen 


154-4 


49-7 


0-43 


Hydrogen 


33-2 


12-8 


0031 


Helium-4 


5-25 


2-26 


0069 


Helium-3 


3-33 


115 


0041 



It has already been mentioned in §52 that solid (crystalline) 
substances differ fundamentally from liquids and gases in being 
anisotropic. The transition between a liquid and a crystal there- 
fore cannot be made in a continuous manner as can that between 
a liquid and a gas. We can always say to which of the two phases 
(crystal or liquid) a body belongs, according to whether it does 
or does not have the qualitative property of anisotropy. Thus 
there cannot exist a critical point for melting. 



210 PHASE TRANSITIONS [IX 

§70. Van der Waals' equation 

As the density of a gas increases, its properties deviate more 
and more from those of an ideal gas, and it finally condenses to 
a liquid. These phenomena depend on complex molecular inter- 
actions, and there is no way of giving a quantitative description 
of these in order to derive theoretically an exact equation of 
state for the substance. We can, however, construct an equation 
of state which takes account of the main qualitative properties 
of molecular interaction. 

The nature of the interaction between molecules has already 
been described in §39. The repulsive forces which rapidly in- 
crease at short distances signify, roughly speaking, that the mole- 
cules occupy a certain definite volume, and the gas cannot be 
compressed beyond this. Another fundamental property of the 
interaction is that there is attraction at large distances; this 
attraction is very important, since it is responsible for the 
condensation of a gas into a liquid. 

First of all, let us take into account, in the equation of state 
(which will be written for one mole of substance), the limited 
compressibility of the gas. To do so, we must replace the volume 
V in the ideal-gas equation p = RT/V by V-b, where b is some 
positive constant which takes into account the size of the 
molecules. The equation 

p =RTI(V-b) 

shows that the volume cannot be made less than b, since for this 
value of V the pressure becomes infinite. 

Let us now take into account the attraction between molecules. 
This attraction must cause a decrease in the gas pressure, since 
each molecule near the wall of the vessel is subject to a force 
towards the interior of the vessel exerted by the other molecules. 
As a rough approximation, this force on each molecule may be 
taken as proportional to the number of molecules per unit volume, 
i.e. to the density of the gas. The pressure itself is also propor- 
tional to this number. Thus the total decrease in pressure due to 
the attraction between molecules is proportional to the square 
of the gas density, i.e. inversely proportional to the square of its 
volume. Accordingly we subtract from the above expression 
for the pressure a term a/V 2 , where a is another constant rep- 



§70] VAN DER WAALS' EQUATION 211 

resenting the forces of molecular attraction. Thus we have the 
equation 

RT a 

P y- b V 2> 



or 



(p + fy(V-b) = RT. 



This is van der Waals' equation. When the gas density is low, 
i.e. the volume V is large, a and b may be neglected, and we 
return to the equation of state of an ideal gas. We shall see below 
that the same equation correctly describes the phenomena which 
occur in the opposite limiting case of high compression. 

To examine the behaviour of a gas described by van der 
Waals' equation, let us consider the isotherms defined by this 
equation, i.e. the curves of p as a function of V for given values 
of T. For this purpose we write the equation in the form 

V p ) p p 

For given values of/? and T this is a cubic equation in V. 

A cubic equation has three roots, of which either all three or 
one may be real; in the latter case the equation also has two 
complex conjugate roots. The volume can, of course, be rep- 
resented as a physical quantity only by real (and positive) 
roots. In the present case the equation cannot have negative 
roots (if p is positive), since if V is negative every term in the 
equation is negative and their sum cannot be zero. Thus we see 
that according to van der Waals' equation there are either three 
different values or one value of the volume corresponding to 
given values of the temperature and pressure. 

The second case always occurs at sufficiently high tempera- 
tures. The corresponding isotherms differ from those of an ideal 
gas only by some change in shape, but remain monotonically 
decreasing (curves 1 and 2 in Fig. 87; increasing numbers on the 
curves correspond to decreasing temperatures). At lower 
temperatures the isotherms have a maximum and a minimum 



212 



PHASE TRANSITIONS 



[IX 



(curves 4, 5, 6), and so for each of them there is a range of 
pressures in which the curve gives three values of V (three 
points of intersection of the isotherm with a horizontal line). 
Figure 88 shows one such isotherm. Let us see what is the 
significance of its various parts. On the sections ge and ca 
the dependence of pressure on volume is of the normal type: 
the pressure increases as the volume decreases. The section 
ec would correspond to the unnatural situation where com- 
pression of the substance would decrease the pressure. It is 




Fig. 87. 

easily seen that such states cannot exist in Nature. For let us 
imagine a substance with these properties, and suppose that a 
small region of it happens to contract, e.g. owing to the fluctua- 
tions described in §64. Then its pressure will decrease also, i.e. 
become less than the pressure of the surrounding medium, which 
in turn causes a further contraction, and so on, i.e. this small 
region will contract at an increasing rate. Thus these states of 
matter would be completely unstable and therefore could not 
occur in reality. 

The existence of the unrealisable section ec of the isotherm 
signifies that, as the volume gradually varies, the substance 
cannot remain homogeneous at all times: at some point there 



§70] 



VAN DER WAALS' EQUATION 



213 

must be a discontinuous change of state and the substance must 
separate into two phases. In other words, the true isotherm is 
given by the curve abfg. The part ab corresponds to the gaseous 
state of the substance, and fg to the liquid state. The straight 
horizontal section bf corresponds to two-phase states where the 
gas becomes a liquid; this occurs at a certain constant pressure 
(for a given temperature). [It can be shown that the section bf 
must be situated so that the areas bed and defare equal.] 




Fig. 88. 

The sections be and ef of the isotherm correspond to meta- 
stable states of supercooled vapour and superheated liquid 
(§68). We now see that there are limits (represented by the 
points c and e) beyond which the vapour cannot be supercooled 
or the liquid superheated. 

As the temperature rises, the straight section of the isotherm 
becomes shorter, and at the critical temperature it contracts to 
a point (K in Fig. 87). The isotherm 3 which passes through 
this point separates isotherms of two types: the monotonic 
isotherms (1,2) and the isotherms with minima and maxima 
(4,5,6), on which the substance must necessarily separate into 
two phases. 

If the beginning and end of the straight section of each iso- 
therm are joined by a curve (a in Fig. 87), this gives the curve 
of phase equilibrium of liquid and vapour in the pV diagram. 
The maximum K on this curve is the critical point. On joining 
the points which correspond to c and e in Fig. 88 we obtain a 
curve (b in Fig. 87) which is the boundary of the region within 
which the substance cannot exist without separation into two 
phases, even in a metastable state. 



214 PHASE TRANSITIONS [IX 

At the critical point the three points at which the straight 
section intersects the van der Waals isotherm merge into one. 
Hence it follows that the tangent to the isotherm at the critical 
point is horizontal, i.e. the derivative of the pressure with 
respect to the volume (at constant temperature) is zero: (dpldV) T 
= 0. The reciprocal of this quantity is the compressibility of 
the substance, which is therefore infinite at the critical point. 

The section of the isotherm which corresponds to a super- 
heated liquid may lie partly below the axis of abscissae (as on 
the isotherm 6 in Fig. 87). This section corresponds to metastable 
states of an "expanded" liquid, as discussed at the end of §5 1 . 

§7 1 . The law of corresponding states 

The critical values of the volume, temperature and pressure 
can be related to the parameters a and b in van der Waals' 
equation. To do so, we note that for T = T c and p = p c all three 
roots of van der Waals' equation 

/ RT C \ a ab 
V 3 -(b + -)V 2 +—V = 

V Pc J Pc Pc 

are the same and equal to the critical volume V c . This equation 
must therefore be identical with 

(Y- V C ) 3 =V 3 -3V 2 V C + 3VV C 2 -V C 3 = 0. 

A comparison of coefficients of powers of V in the two equations 
gives the three relations 

RT r a ab 

b + —±=3V c , ~=3V C \ -=V C \ 

Pc Pc Pc 

These relations, regarded as equations for the unknowns V c , p c 
and T c , are easily solved to give 

V c = 3b, p c = a/21b 2 , T c = $a/21bR. 

By means of these relations we can carry out the following 
interesting transformation of van der Waals' equation. In this 
equation we write, instead of the variables p, T, V, their ratios 
to the critical values: 



§71] THE LAW OF CORRESPONDING STATES 215 

p*=plp c , T* = T/T c , V* = V/V c ; 

these ratios are called the reduced pressure, temperature and 
volume. By means of simple transformations we can easily see 
that van der Waals' equation then becomes 



3 

V : 



P*+17«)(3^*-l) = 8r 



The precise form of this equation is not of particular interest; 
the remarkable thing is that it does not involve the constants 
a and b which depend on the nature of the gas. In other words, 
if the critical values are used as the units of measurement of 
the volume, pressure and temperature, the equation of state 
becomes the same for all substances. This is called the law of 
corresponding states. 

If this law applies to the equation of state, it will apply also to 
all phenomena which are in any way related to the equation of 
state, including the gas-liquid phase transition. For example, 
the temperature dependence of the saturated vapour pressure, 
if written as a relation between reduced quantities p/p c =f(T/T c ), 
must be a universal relation. 

A similar conclusion may be drawn concerning the heat of 
evaporation q. Here we must consider the dimensionless ratio 
of the heat of evaporation to some other quantity of the same 
dimensions (energy/mole); this may be taken as RT C . According 
to the law of corresponding states, the ratio qlRT c must be the 
same for all substances as a function of the reduced temperature: 
qlRT c = F(T/T C ). For temperatures much below the critical 
temperature this function tends to a constant limit, whose 
experimental value is about 10. 

It should be emphasised that the law of corresponding states 
is only approximate, but it can be used to derive results which are 
entirely suitable for rough estimations. 

Although the law of corresponding states has been derived from 
van der Waals' equation, it is in fact somewhat more accurate 
than the latter, since it does not depend on the specific form of 
the equation of state, but follows simply from the fact that this 
equation involves only two constants a and b. A different equation 
of state with two parameters would likewise lead to the law of 
corresponding states. 



216 



PHASE TRANSITIONS 



[IX 



§72. The triple point 

As we know, equilibrium between two phases is possible only 
when a certain relation holds between the temperature and the 
pressure, represented by a certain curve in the pT plane. It is 
evident that three phases of the same substance cannot be 
simultaneously in equilibrium with one another along a line; such 
an equilibrium is possible only at a particular point in the pT 
diagram, i.e. at a particular pressure and a particular temperature. 
This is the point at which the equilibrium curves of each pair of 
the three phases intersect. Points of equilibrium of three phases 
are called triple points. For example, for water the simultaneous 
existence of ice, steam and liquid water is possible only at 
4-62 mm Hg pressure and +0-01°C temperature. 

Since even three phases are in equilibrium only at one point, 
four or more phases cannot exist simultaneously in equilibrium 
with one another. 

The fact that triple points correspond to definite values of the 
temperature makes them especially suitable as fixed points of 
the temperature scale. Their reproduction is free from the 
difficulties associated with the need to maintain a given pressure, 
as is required, for example, when the melting point of ice at 
atmospheric pressure (or any point of equilibrium of two phases) 
is taken as a fixed point. The precise definition now used for the 
absolute degree is based on such a choice: the temperature of 
the triple point of water is taken to be exactly 273-16°K. It 
should be mentioned, however, that with the present accuracy 
in the measurement of temperature and pressure this definition 
is indistinguishable from that in which the melting point of ice 
is taken as 273 -15°K. 




Fig. 89. 



§72] 



THE TRIPLE POINT 



217 



Figure 89 shows the form of the phase diagram for a substance 
having only three phases: solid, liquid and gaseous. In the 
diagram, the regions marked s, I and g correspond to these 
phases, and the lines separating the regions are the curves of 
equilibrium of the corresponding pairs of phases. The direction 
of the melting curve is chosen so as to correspond to the usual 
case where a body expands on melting (see §67). For the few 
instances where melting is accompanied by contraction of the 
substance, the curve slopes in the opposite direction. 

It is seen from the phase diagram that a substance does not 
necessarily pass through a liquid state in the course of becoming 
a gas. At pressures below the triple point, heating the solid 
converts it directly into vapour; this phase transition is called 
sublimation. For example, solid carbon dioxide sublimes at 
atmospheric pressure, since its triple point corresponds to a 
pressure of 5- 1 atm (and a temperature of — 56-6°C). 

The curve of equilibrium of a liquid and a gas terminates at the 
critical point (K in Fig. 89). For transitions between liquid and 
solid phases there can be no critical point (as already mentioned 
in §69). The melting curve therefore cannot simply terminate, 
and must continue indefinitely. 

The curve of equilibrium of a solid and a gas passes through 
the origin, i.e. at absolute zero temperature a substance is in the 
solid state at any pressure. This is a necessary consequence of 
the ordinary concept of temperature based on classical mechanics. 
According to this concept, the kinetic energy of the atoms is 
zero at absolute zero temperature, i.e. all the atoms are at rest. 
The equilibrium state of a body is then one in which the configur- 
ation of the atoms corresponds to the minimum energy of 
interaction between the atoms. This configuration, whose 
properties differ from all others, must have some degree of order- 



25 at 




218 PHASE TRANSITIONS [IX 

ing, i.e., must represent a spatial lattice. This means that the 
substance must be crystalline at absolute zero. 

There exists in Nature, however, one exception to this rule: 
helium, after becoming liquid, remains liquid at all temperatures 
down to absolute zero. The phase diagram of the isotope helium-4 
is shown in Fig. 90; the broken line in this diagram will be 
explained in §74. We see that the evaporation and melting curves 
nowhere intersect, i.e. there is no triple point. The melting curve 
meets the ordinate axis at p = 25 atm; this means that, in order 
to solidify helium, it must be not only cooled but at the same 
time subjected to a pressure of at least 25 atm. 

It is clear from the above that this behaviour of helium is 
inexplicable on the basis of classical ideas; it is in fact due to 
quantum effects. As already mentioned in §50, according to 
quantum mechanics the motion of the atoms does not cease 
entirely even at absolute zero. For this reason the above con- 
clusion that a substance must solidify at this temperature is 
also incorrect. The quantum properties of a substance appear 
more markedly at low temperatures, where they are not masked 
by the thermal motion of the atoms. All substances except 
helium solidify before their quantum properties become suffi- 
ciently important; only helium becomes a "quantum liquid" 
which need not solidify. Other remarkable properties of this 
liquid will be discussed in § 1 24. 

§73. Crystal modifications 

The region of the solid state is not usually occupied by a 
single phase. At different pressures and temperatures a substance 
may be in different crystal states, each with a definite structure. 
These different states are also different phases of the substance, 
and are called crystal modifications; the property of having more 
than one such modification is very common, and is called poly- 
morphism of the substance (or, for the elements, allotropy). Well- 
known examples are the modifications of carbon (graphite and 
diamond), sulphur (which forms orthorhombic and monoclinic 
crystals), and silica (the various minerals quartz, tridymite, cris- 
tobalite). 

Like any other phases, different modifications can be in equilib- 
rium with one another only along certain lines in the pT diagram; 
the transition from one modification to another (called a poly- 



§73] 



CRYSTAL MODIFICATIONS 



219 



morphic transformation) is accompanied by absorption or 
evolution of heat. For example, the transformation of what is 
called a iron (with a body-centred cubic lattice) into y iron (with 
a face-centred cubic lattice) occurs at 910°C at atmospheric 
pressure, and is accompanied by the absorption of about 1600 
J/moleofheat. 

Figure 91 shows, as an example, the general form of the phase 
diagram for sulphur. The letters R and M denote the regions 
where the two solid phases are stable: rhombic (ordinary yellow 
sulphur) and monoclinic. We see that here there are three triple 
points. 

Pa 




Liquid 



Fig. 91. 

Figure 92 shows the phase diagram of water. The five crystal 
modifications of ice are shown in the diagram by the numbers 
I, II, III, V, VI. Ordinary ice corresponds to region I; the other 
modifications are formed only under pressures of thousands of 
atmospheres. The vapour region corresponds to such low 
pressures that it is almost impossible to show it in the same 
diagram. 




-40 -20 20 



Fig. 92. 



T, °C 



220 PHASE TRANSITIONS [iX 

A typical feature of phase transitions between different crystal 
modifications is the ease with which metastable states can occur. 
Supercooling of a vapour or superheating of a liquid is possible 
only when the necessary precautions are taken, but the delay 
of phase transitions in the solid state and existence of crystal 
modifications in conditions where they are "not permitted" are 
almost the rule. This is quite understandable, since the closeness 
of atoms in a crystal and the restriction of their thermal motion 
to small oscillations greatly hamper the rearrangement of the 
lattice into a different modification. An increase in temperature 
makes the thermal motion more violent and thus accelerates this 
rearrangement. 

Here it should be remembered that the polycrystalline structure 
of a solid is itself in a sense metastable (in comparison with the 
single-crystal state). Thus, when a body composed of small 
crystals is heated, its component crystals become larger, some 
crystals growing at the expense of others; this is called recrystal- 
lisation. The amorphous state of a body may also be metastable; 
for example, spontaneous crystallisation is the reason for the 
cloudiness of very old glass. 

A polymorphic transformation is facilitated by the presence 
in the former phase of inclusions of the new phase, which act 
as "nuclei". A well-known example of this is the transformation 
of ordinary white tin (which has a tetragonal structure) into a 
powder of grey tin (a modification which has a cubic lattice). At 
atmospheric pressure these modifications are in equilibrium at 
18°C, white tin being stable above this temperature and grey 
tin below. In practice, however, white tin can exist even below 
freezing point, but when a few grains of the grey modification 
are added it crumbles to a grey powder. 

The difficulty of lattice rearrangement at low temperatures may 
bring about the existence of modifications which are not stable 
phases under any conditions; such modifications do not appear 
at all in the phase diagram, which represents stable states of a 
substance. This is observed, for example, in the hardening of 
steel. The solid solution of carbon in y iron, called austenite, 
is stable only at temperatures 700-900°C (depending on the 
carbon content), and must decompose at lower temperatures. 
When austenite is very rapidly cooled or quenched, however, 
there occurs instead a formation in the metal of needle-shaped 



§73] CRYSTAL MODIFICATIONS 221 

crystals of a new phase, a solid solution with a tetragonal lattice 
called martensite, which is extremely hard. This "intermediate" 
phase is always metastable, and decomposes when steel is slowly 
heated or tempered at 250-300°. 

Figure 93 shows the phase diagram of carbon; the region of 
the gaseous state is at low pressures and is not visible on the 
scale used in this diagram. The phase diagram shows that at 
ordinary pressures and temperatures the stable modification is 
graphite. But under ordinary conditions graphite and diamond 
both exist as almost completely stable crystals. This is due to 



200 


i 


160 


\ 




Diamond I 


120 


^r \ Liquid 


80 




40 


^r Graphite / 

i i i i/ i i 





2000 4000 6000 




T, °K 




Fig. 93. 



the large difference in the structures of the two crystals, which 
requires a very extensive rearrangement to convert one into 
the other (as is indicated by the fact that the density of diamond 
is 1-5 times that of graphite). On heating to high temperatures, 
however, diamond is transformed into graphite: above 1700°K 
it rapidly crumbles into graphite powder (if heated in a vacuum 
to prevent combustion). The diagram shows that the reverse 
process of conversion of graphite into diamond can occur only 
at very high pressures. The region in which diamond is stable 
lies above about 10 000 atm. A high temperature is also necessary 
if the process is to occur at a reasonable rate. The process is 
carried out in practice at pressures of 50 000 to 100 000 atm and 
temperatures of 1 500-3 000°K, and a metal catalyst must also 
be present. The spontaneous transformation of graphite into 
diamond has been observed at about 130 000 atm at temperatures 
above 3300°K; this is apparently in the region where graphite 
is neither stable nor metastable, but completely unstable. 



222 PHASE TRANSITIONS [IX 

§74. Phase transitions of the second kind 

It has already been mentioned that the transition between 
phases of different symmetry cannot occur in crystals in a con- 
tinuous manner as it can in a liquid and a gas. In every state the 
body has one symmetry or the other, and therefore we can 
always assign it to one of the two phases. 

The transition between different crystal modifications is usually 
effected by means of a phase transition in which there is a sudden 
rearrangement of the crystal lattice and the state of the body 
changes discontinuously. As well as such discontinuous tran- 
sitions, however, another type of transition involving a change 
of symmetry is also possible. 

Actual examples of such transitions are somewhat complicated 
as regards the details of the crystal structure of the bodies. To 
illustrate the nature of these transitions, we shall therefore 
consider an imaginary example. Let us suppose that a body 
crystallises at low temperatures in the tetragonal system, i.e. 
has a lattice consisting of cells which are rectangular paral- 
lelepipeds with square bases and a height c greater than the side 
a of the base. Let the difference between a and c be small, i.e. 
the crystal be tetragonal but almost cubic; and let us assume 
that during thermal expansion the a edges increase in length more 
rapidly than the height c. Then, as the temperature increases, 
the lengths of the sides of the unit parallelepiped will become 
more nearly equal and at a certain temperature they will be the 
same; on further heating, all three sides will increase in length 
at the same rate, remaining equal. It is clear that, as soon as 
a becomes equal to c, the symmetry of the lattice suddenly 
changes from tetragonal to cubic, and we have essentially a 
different modification of the substance. 

This example is typical in that there is no discontinuous change 
in state of the body. The configuration of the atoms in the crystal 
changes continuously. However, an arbitrarily small displace- 
ment of the atoms from their symmetrical position in the lattice 
of the cubic modification (when the temperature falls again) is 
sufficient to cause a sudden change in the symmetry of the lattice. 
So long as all three sides of the cell are equal the lattice is cubic, 
but the appearance of even an infinitesimal difference between 
the lengths a and c makes the lattice tetragonal. 

A transition between crystal modifications which occurs in 



§74] PHASE TRANSITIONS OF THE SECOND KIND 223 

this way is called a. phase transition of the second kind, in contrast 
to ordinary phase transitions, which in this case are said to be of 
the first kind, t 

Thus a phase transition of the second kind is continuous in 
the sense that the state of the body changes continuously. It 
should be emphasised, however, that the symmetry at the tran- 
sition point does, of course, change discontinuously, so that 
we can always assign the body to one of the two phases. But, 
whereas at a phase transition point of the first kind bodies in two 
different states are in equilibrium, at a transition point of the 
second kind the states of the two phases are the same. 

The absence of a discontinuity of state in a phase transition 
of the second kind means that there is no discontinuity in quantities 
which describe the thermal state of a body: volume, internal 
energy, enthalpy, etc. Hence, in particular, such a transition is 
not accompanied by evolution or absorption of heat. 

Nevertheless, at a transition point there is a discontinuous 
change in the dependence of these quantities on temperature. 
For instance, in the example considered it is evident that the 
thermal expansion of the crystal will occur differently according 
as there is only a change in the volume of the lattice (when the 
crystal has cubic symmetry) or the heating also brings about a 
change in cell shape as a result of unequal changes in the height 
and base edge of the cells, as when there is tetragonal symmetry. 
It is also evident that different quantities of heat will be necessary 
for the same temperature increase. 

This means that at a transition point of the second kind there is 
a discontinuity in the temperature derivatives of the thermal 
properties of the body, i.e. in the coefficient of thermal expansion 
(dV/dT) p , the specific heat C p = (dW/dT) p , etc. 

The presence of these discontinuities is the main characteristic 
of transitions of the second kind which appears in thermal 
measurements. Figure 94 shows the typical manner of variation 
of specific heat with temperature near a transition point of this 

t The example described above is not entirely imaginary. A change of this 
type occurs in the lattice of barium titanate (BaTi0 3 ). At room temperature 
this lattice is tetragonal, with values of a and c which differ by 1%. When the 
temperature increases, the length a increases and c decreases. At 120°C the 
substance changes to the cubic modification, but in this actual case the values of 
a and c do in fact have a slight discontinuity at the transition point, so that the 
transition is of the first kind. 



224 



PHASE TRANSITIONS 



[IX 



kind: a gradual increase is interrupted by a sudden drop, after 
which the specific heat again begins to rise. 

In a transition of the second kind, the pressure derivatives of 
thermal quantities are also discontinuous. For example, the 
compressibility (dVldp) T has a discontinuity. 

Let us return to the imaginary example used above. The follow- 
ing property of the change in symmetry in the transition may be 
noted: the lattice of the cubic modification has all the symmetry 
of the elements of the tetragonal modification and some other 
elements as well. In this sense we can say that the transition takes 
place between two phases of which one has higher symmetry 
than the other. This is in fact a general property and applies to 
all phase transitions of the second kind. 



Fig. 94. 



This is a limitation (in reality not the only one) on the possible 
existence of a phase transition of the second kind. For example, 
no such transition can occur between crystals of the cubic and 
hexagonal systems: neither of these symmetries can be said to 
be higher than the other, since the former contains axes of order 
4 which do not occur in the latter, but it does not, on the other 
hand, contain the axis of order 6. 

It can also be shown that a transition of the second kind cannot 
occur between a crystal and a liquid. 

The direction in which the specific heat changes discontinuously 
at a transition of the second kind is related to the way in which 
the symmetry changes: the specific heat is smaller in the phase 
of higher symmetry. In most cases the high-temperature phase 
has the higher symmetry, and the discontinuity of the specific 
heat is then as shown in Fig. 94. This sequence of the phases 
with respect to temperature is not necessary, however. For 
example, Rochelle salt (NaK(C 4 H 4 6 ).4H 2 0) has two transition 



§75] ORDERING OF CRYSTALS 225 

points of the second kind (at - 18 and -23°C), between which 
its crystals belong to the monoclinic system; at temperatures 
outside this range the salt forms orthorhombic crystals. It is 
clear that the passage through the upper point in the direction 
of increasing temperature is accompanied by an increase of 
symmetry, but the passage through the lower point involves a 
corresponding decrease of symmetry. 

It has already been mentioned that ordinary phase transitions 
frequently exhibit phenomena of superheating or supercooling, 
in which one phase continues to exist (as a metastable phase) 
under conditions where the other phase is stable. The nature 
of these phenomena depends on the necessity for "centres" on 
which the new phase can grow. In transitions of the second kind, 
such phenomena are obviously impossible, since one phase 
changes into the other instantaneously and continuously. This 
is very clearly seen in the example considered above, where 
the transition amounted essentially to a change in the configuration 
of the atoms in thermal expansion. 

Phase transitions of the second kind are not always transitions 
between different crystal modifications, but they always bring 
about some new qualitative property of a body, with a continuous 
change of state. This may be a new symmetry property (related 
to the magnetic properties of the substance), or it may be the 
occurrence of what is called superconductivity, the disappearance 
of electrical resistance. 

Finally, there is a very unusual phase transition of the second 
kind in liquid helium at about 2-2°K. In this transition the liquid 
remains a liquid but acquires fundamentally new properties 
(see §124). The broken line in the helium phase diagram (Fig. 
90, §72) divides the regions of existence of the two phases, 
which are known as helium I and helium II. 

§75. Ordering of crystals 

All the crystal structures discussed in §47 have the property 
that the atoms of each kind are situated at entirely definite 
positions and, conversely, at each lattice point there must be 
an atom of a particular kind. The number of atoms of each kind is 
equal to the number of places for them in the lattice. 

There are also structures which do not have this property, 
however; for example, that of sodium nitrate (NaNO s ). We shall 



226 PHASE TRANSITIONS [IX 

not describe this in detail, but simply mention that in this crystal 
the N0 3 groups form layers in which the nitrogen atoms are at 
the vertices of equilateral triangles and the oxygen atoms surround 
the nitrogen atoms in position a or b (Fig. 95). The possibility 
of these two orientations of the N0 3 groups implies that the 
number of positions which can be occupied by oxygen atoms is 
twice the number of these atoms. 



000 



: (a) 

-6.. 


O O 





Y(b) 




• 


ON 


•0 


Fig. 


95. 



At sufficiently low temperatures, the oxygen atoms take up 
quite definite positions; in practice, what happens is that in each 
layer all the N0 3 groups have the same orientation, and layers 
with orientation a alternate with those having orientation b. 
Such a crystal is said to be completely ordered. 

When the temperature is raised, however, the ordered arrange- 
ment of the atoms is disturbed: as well as N0 3 groups occupying 
the usual (their "own") position, there appear groups with the 
"other" orientation. 

As the degree of ordering decreases, i.e. as the fraction of 
"incorrectly" oriented NG 3 groups increases, a point is finally 
reached, at a temperature of 275°C, where the "own" and "other" 
orientations are entirely mixed: each N0 3 group has an equal 
probability of occupying either position. The crystal is then said 
to be disordered. All the N0 3 layers become crystallographically 
equivalent, i.e. there is a change, namely an increase, in the 
symmetry of the crystal. 

Phenomena of crystal ordering occur very widely in alloys. 
For example, crystals of brass (the alloy CuZn) at low tempera- 
tures have a cubic lattice with the copper atoms at the vertices 
and the zinc atoms at the centres of the cubic cells (Fig. 96a). 
This structure corresponds to a completely ordered crystal. 



§76] 



LIQUID CRYSTALS 



227 



The copper and zinc atoms may change places, however; in 
this sense we may say that in this crystal also the number of 
places available to atoms of each kind is twice the number of 
those atoms. As the temperature increases, the number of "in- 
correctly" placed atoms becomes larger, and complete disorder 
exists at 450°C: at each lattice point a copper atom or a zinc atom 
can occur with equal probability, so that all the lattice points 
become equivalent (Fig. 96b). At this stage the symmetry of the 
crystal obviously changes: its Bravais lattice becomes body- 
centred cubic instead of simple cubic. 



X 



(a) xZn oCu (b) 

Fig. 96. 

In both the examples described above, the transition to the 
disordered state occurs by a phase transition of the second kind. 
The degree of ordering decreases continuously and becomes zero 
at a certain temperature, which is the transition point. 

This type of transition to the disordered state is not a general 
rule, however; the change can also occur by an ordinary dis- 
continuous phase transition. In such cases the ordered con- 
figuration of the atoms in the crystal is destroyed at first only to 
a comparatively small extent as the temperature increases, and 
at a certain temperature the crystal suddenly enters the dis- 
ordered state, in which the atoms are completely intermingled. 
Such a transition occurs, for example, at 390°C in the alloy 
Cu 3 Au. In its disordered phase the copper and gold atoms are 
randomly located at all the points of a face-centred cubic lattice; 
in the ordered crystal the gold atoms occupy positions at the 
vertices of the cubic cells, and the copper atoms at the centres 
of the faces. 



§76. Liquid crystals 

In addition to the anisotropic crystalline and isotropic liquid 
states, a substance may also exist in a peculiar state called a 



228 PHASE TRANSITIONS [IX 

liquid crystal. In its mechanical properties, a substance in this 
state resembles an ordinary liquid, being fluid; liquid crystals 
include substances both of high mobility (low viscosity) and of 
low mobility (high viscosity). These liquids nevertheless differ 
from ordinary liquids in being anisotropic; this is most noticeable 
in their optical properties. 

The liquid-crystal state is observed in many complex organic 
substances having large molecules, usually of elongated form. 
It is not uncommon; about one complex organic substance out 
of every two hundred forms liquid crystals. 

The physical nature of the liquid-crystal state appears to be 
as follows. In an ordinary liquid the relative position and orienta- 
tion of the molecules are completely random; in other words, 
the molecules of a liquid in their thermal motion undergo both 
random translational movements and random rotations. In a 
liquid crystal, however, although the molecules are randomly 
situated in space, their mutual orientation is not random. In 
other words, only the translational thermal motion of the mole- 
cules is random, and not their rotation. The simplest example of 
such a structure can be imagined as a liquid consisting of rod- 
shaped molecules, which can move in any manner relative to 
one another provided that they remain parallel. Since there is 
no obstacle to the translational motion of the molecules, the 
substance is fluid, i.e. behaves as a liquid, but the ordered 
arrangement of the molecules has the result that the substance 
is anisotropic. For instance, it is clear that the properties of the 
substance in the direction parallel to the rod-shaped molecules 
will be entirely different from its properties in other directions. 

A substance in the liquid-crystal state is not usually a "single 
crystal", but forms a"polycrystalline" mass consisting of a large 
number of droplet-like liquid crystallites oriented variously 
with respect to one another. For this reason, a substance which 
is a liquid crystal usually has the appearance of a turbid liquid: 
this occurs because of the random scattering of light at the 
boundaries between droplets. By means of a strong electric or 
magnetic field it is possible in some cases to give all the droplets 
the same orientation, and an almost clear liquid "single crystal" 
is obtained. 

If a liquid crystal is placed in a liquid with which it does not 
mix, the individual liquid-crystal drops take a form which is 



§76] LIQUID CRYSTALS 229 

sometimes spherical, sometimes ellipsoidal, and in a few cases 
even that of strange polyhedra with much rounded edges and 
corners. 

Substances which exist in the liquid-crystal state also possess 
ordinary solid-crystal and isotropic liquid phases. The sequence 
of formation of these phases is as follows. At low temperatures 
the substance is a solid crystal, at higher temperatures it enters 
the liquid-crystal state, and at still higher temperatures it becomes 
an ordinary liquid. Many substances form not only one but two or 
more different liquid-crystal modifications. Like all phase 
transitions, the transformations of liquid-crystal phases into 
one another or into other phases occur at precisely defined 
temperatures and are accompanied by the evolution or absorption 
of heat. 



CHAPTER X 

SOLUTIONS 



§77. Solubility 

Solutions are mixtures of two or more substances in which the 
substances are mixed on the molecular scale. The relative amounts 
of the various substances in the mixture may vary over a more 
or less wide range. If one substance is present in greater quantity 
than the others, it is called the solvent, and the other substances 
are called solutes. 

The composition of a solution is described by its concentration, 
which gives the relation between the quantities of the substances 
in the mixture — the components of the mixture, as they are called. 
The concentration can be defined in various ways. Physically, 
the most informative is the molar concentration, i.e. the ratio 
of the numbers of molecules (or, what is the same thing, the 
ratio of the quantities expressed in moles). Alternatively, we may 
use concentrations by weight, by volume (the volume of substance 
dissolved in a given volume of solvent), and so on. 

The process of dissolution is accompanied by the evolution or 
absorption of heat. The quantity of heat depends not only on the 
quantity of solute but also on the quantity of solvent. 

The heat of solution is usually defined as the quantity of heat 
evolved or absorbed in the dissolution of one gram-molecule of 
substance in a quantity of solvent so large that any further dilution 
would cause no thermal effect. For example, the heat of solution 
of sulphuric acid (H 2 S0 4 ) in water is +75 000 J (the plus sign 
denoting that heat is evolved); the heat of solution of ammonium 
chloride (NH 4 C1) is —16 500 J (the minus sign shows that heat 
is absorbed). 

The mutual solubility of two substances usually has definite 
limits: no more than a certain amount of solute can dissolve in 
a given quantity of solvent. A solution containing the maximum 
possible quantity of solute is said to be saturated. If further 

230 



§77] SOLUBILITY 23 1 

solute is added to such a solution, it will not dissolve, and we 
can therefore say that a saturated solution is one which is in 
thermal equilibrium with the pure solute. 

The concentration of the saturated solution is a measure of the 
ability of a given substance to dissolve in the solvent concerned, 
and is also called simply the solubility of the substance. 

The solubility in general depends on the temperature. By 
means of Le Chatelier's principle we may relate the nature of 
this dependence to the sign of the heat of solution. Let us suppose 
that dissolution is accompanied by absorption of heat (as when 
ammonium chloride is dissolved in water), and that we have a 
saturated solution in equilibrium with undissolved solid. If 
this system is heated it will no longer be in equilibrium, and 
processes must occur which tend to oppose the external inter- 
action (heating) which has brought the substance out of equilib- 
rium. In the present case this means that the solubility of the 
substance in water will increase so as to allow further dissolution, 
accompanied by absorption of heat. 

Thus, if dissolution is accompanied by absorption of heat, 
the solubility increases with temperature, but if heat is evolved 
on dissolution an increase in temperature will cause a decrease 
in solubility. 

The dissolution of a gas in a liquid is usually accompanied by 
a large decrease in volume: the volume of the solution is con- 
siderably less than the sum of the original volumes of the solvent 
and the dissolved gas (for example, when one mole of nitrogen is 
dissolved in a large quantity of water at room temperature and 
atmospheric pressure the volume of the liquid increases by only 
40 cm 3 , whereas the volume of this amount of gas is 22 400 cm 3 ). 
Hence it follows, by Le Chatelier's principle, that the solubility 
of a gas in a liquid increases with the gas pressure over the liquid, 
at a given temperature. 

The way in which the solubility of a gas depends on its pressure 
is easily established for weak solutions. [Weak (or dilute) 
solutions are those in which the number of solute molecules is 
small in comparison with the number of solvent molecules.] 
For this purpose we use the fact that thermal equilibrium (in 
this case, equilibrium between the gas and its saturated solution) 
is a dynamic equilibrium on a molecular scale. This means that, 
after equilibrium has been reached, the gas molecules pass from 



232 SOLUTIONS [x 

the gas to the solution and back, but the number of molecules 
entering the solution from the gas per unit time is equal, in 
equilibrium, to the number of gas molecules leaving the solution 
per unit time. The number of gas molecules entering the liquid is 
proportional to the number of collisions per unit time between gas 
molecules and the liquid surface. This number in turn is propor- 
tional to the density of the gas (at a given temperature), and 
therefore to its pressure. Similarly, the number of gas molecules 
leaving the solution is proportional to its concentration. Thus, 
from the equality of the two numbers, it follows that the con- 
centration of a saturated solution, i.e. the solubility of the 
gas, is proportional to the gas pressure over the solution 
(Henry's law). 

It should be mentioned that this law is valid only for a weak 
solution, since in other solutions the foregoing arguments are 
invalid on account of the interaction between gas molecules in 
the solution. In consequence of this interaction the number of 
these molecules leaving the solution can no longer be assumed to 
be simply proportional to the concentration. Henry's law is 
therefore applicable, for instance, to oxygen and nitrogen, whose 
solubility in water is low, but not to the dissolution of carbon 
dioxide or ammonia, which are readily soluble in water. 

In the great majority of cases, the dissolution of a gas is 
accompanied by evolution of heat; this is a quite natural result of 
the passage of molecules from a gaseous medium, where the inter- 
action between molecules is weak, to a medium where the gas 
molecules are subject to a strong attraction exerted by solvent 
molecules. For this reason the solubility of gases in liquids 
decreases with increasing temperature (at a given pressure). 

§78. Mixtures of liquids 

Substances which are so rarefied that the interaction between 
their molecules is unimportant can mix freely with one another. 
In this sense we may say that all gases mix in any proportions. 

In the mixing of liquids, however, various cases can occur. 
There exist liquids which mix in any proportions, for example 
alcohol and water, but the mutual solubility of other liquids is 
limited to various extents. For instance, water and paraffin are 
almost insoluble in each other; not more than 8% (by weight) 
of ether can be dissolved in water at room temperature, and so on. 



§78] 



MIXTURES OF LIQUIDS 



233 



The mutual solubility properties of liquids can be conveniently 
represented graphically by plotting as abscissa the concentration 
c of the mixture (e.g. in percent by weight) and as ordinate the 
temperature (or the pressure, if we are considering the dependence 
of solubility on pressure at a given temperature). 

Figure 97 shows a diagram of this kind for a mixture of water 
and phenol (C 6 H 5 OH). One of the vertical axes corresponds to 
0% water, i.e. pure phenol, and the other to pure water. 




Fig. 97. 



All points outside the hatched region of the diagram correspond 
to homogeneous mixtures of the two components; the curve 
forming the boundary of the hatched region represents the limit 
of their miscibility. For example, at the temperature corres- 
ponding to the horizontal line ae, the point b gives the limiting 
solubility of water in phenol, and the point d that of phenol in 
water. If water and phenol are mixed in quantities corresponding 
to a point c within the hatched region, the liquid separates into 
two horizontal layers with the denser layer below and the less 
dense one above. These two liquid layers which coexist in equilib- 
rium represent two different phases. One is a saturated solution 
of water in phenol (represented by the point b), and the other is 
a saturated solution of phenol in water (the point d). It is easily 
shown, in exactly the same way as in §66 for the liquid- vapour 
phase diagram, that the quantities of the two phases will again 
be determined by the lever rule: they are inversely proportional to 
the lengths of cb and cd. 

If the mutual solubility of two liquids increases with tempera- 
ture, a point may be reached at which their miscibility becomes 
unrestricted. This occurs, for example, with phenol and water: at 
temperatures above 70°C, the two liquids mix in any proportions. 



234 



SOLUTIONS 



[x 



This limiting temperature is called the critical temperature 
of mixing, and the corresponding point K in the phase diagram 
(Fig. 97) is called the critical point of mixing; the properties of 
this point are in many ways similar to the critical point in the 
equilibrium between a liquid and a gas. 



T 






________ 




























y — : ^ 




^_ ~_7 


I8-5°C 





100% 
triethylamine 

Fig. 98. 



100% 
water 




60-8 



Fig. 99. 



There are also cases where the critical point is not the upper 
but the lower limit of the region of restricted miscibility of two 
liquids, for example water and triethylamine (N(C 2 H 5 ) 3 ), which 
mix in any proportions at temperatures below a certain critical 
temperature (Fig. 98). Finally, in some cases there are two 
critical temperatures, an upper and a lower, between which the 
mutual solubility of the two liquids is restricted. This occurs, 
for instance, with water and nicotine (Fig. 99). 

§79. Solid solutions 

Some substances are capable of forming crystals containing 
the atoms of two different substances. These are called solid 
solutions or mixed crystals. The ability to form solid solutions is 
especially common among metals, which form alloys with one 
another. 

Mixed crystals may be referred to as solid solutions because 
the composition of the crystals can vary over a more or less wide 
range, whereas for crystals which are "chemical compounds" 
the composition must be entirely definite. The crystal structure 
of a solid solution is directly related to that of one or other 
component, but a chemical compound has a structure of its own. 

The great majority of solid solutions are of what is called the 
substitution type. Such a solution is obtained by replacing some 



§79] SOLID SOLUTIONS 235 

of the atoms in the crystal lattice of one substance by atoms of 
the other substance. For such a replacement to be possible, the 
atoms of the new substance must of course be of about the same 
size as those of the solvent. Substitution-type solutions include, 
in particular, the majority of metal alloys. There are even cases 
of unrestricted mutual solubility of the two components of an 
alloy (for example, alloys of copper and gold); for this to be so it 
is evidently necessary that the two components should have 
crystal lattices of the same type. 

Solid solutions of the substitution type can be formed not only 
by elements but also by chemical compounds, in which case the 
phenomenon is called isomorphism. In such mixed crystals the 
atoms of one compound are replaced by atoms belonging to 
the other compound. 

It is not necessary that the two compounds should be chemi- 
cally similar in order to form solid solutions. The molecular 
structure of the two substances must be of the same type, how- 
ever. Thus, as well as chemically similar isomorphous substances 
(such as ZnS0 4 and MgS0 4 ), we also find pairs of isomorphous 
substances which are chemically not at all similar: BaS0 4 and 
KMn0 4 , PbS and NaBr, etc. 

For isomorphism, not only must the molecular structure be 
of the same type, but the crystal lattices must also be of the same 
type and have similar dimensions. For example, the significance 
of the dimensions is seen from the compounds KC1, KBr and KI, 
all of which have lattices of the same type (NaCl type) but with 
different distances between adjoining atoms (3-14, 3-29 and 3-52 
A respectively). The comparatively small difference between 
the KC1 and KBr lattices enables these compounds to form 
solid solutions of any composition, but the larger difference 
between KC1 and KI has the result that their mutual solubility 
is restricted. An even greater difference may entirely prevent 
isomorphism. 

Another type of solid solution is the interstitial type. In these 
crystals the solute atoms penetrate between the solvent atoms, 
slightly increasing the distance between them. In other words, 
they occupy positions in the lattice which are not occupied in 
the pure solvent. Such solid solutions can, of course, exist only if 
the atoms of the solute are considerably smaller than those of 
the solvent. 



236 



SOLUTIONS 



[X 



Solid solutions of the interstitial type are formed, for example, 
by hydrogen, nitrogen and carbon in certain metals. For instance, 
carbon can dissolve (at high temperatures) in what is called y iron, 
a modification of iron with a face-centred cubic structure; in the 
resulting solution, called austenite, the carbon atoms occupy 
positions at the midpoints of the edges of the cubic cells, between 
the iron atoms at the vertices and face centres of these cells. 
Up to about 10% of such positions can be occupied. 

§80. Osmotic pressure 

If two solutions of different concentrations are separated by 
a porous partition, then both the solvent and the solute will pass 
through the partition until the two solutions are completely 
mixed. There are other partitions, however, which have selective 
transmission, i.e. allow some substances to pass through but not 
others; they are said to be semipermeable. These include various 
animal and vegetable membranes, colloidal films, and partitions 
of porous clay or porcelain in which the pores are closed by films 
of copper ferrocyanide (Cu 2 Fe(CN) 6 ). All these transmit water 
but retain substances dissolved in it. The passage of the solvent 
through such a partition is called osmosis. 

p 

1 



I 



-Water- '// .Solut^orv 



Fig. 100. 

If two vessels are separated by a semipermeable partition (P 
in Fig. 100), and one vessel contains a solution of sugar, say, in 
water, and the other contains pure water, then water is found to 
enter the vessel containing the solution; the solution as it were 
attracts the solvent. This will continue until a certain difference 
in level is established between the water and the solution. The 
pressures in the two vessels are then unequal: in the vessel con- 
taining solution there is an excess pressure equal to the hydro- 
static pressure of the extra column of liquid in that vessel. This 
excess pressure is called the osmotic pressure of the solution. 



§80] OSMOTIC PRESSURE 237 

The reason for this phenomenon is easily understood. Since 
only water can pass through the semipermeable partition, equilib- 
rium of the liquids in the two vessels does not require equality 
of the total pressures on the two sides of the partition. Equilib- 
rium is reached, roughly speaking, when the pressure in the 
vessel containing pure water becomes equal to that part of the 
pressure of the solution which is due to the water molecules. The 
total pressure in the solution will then exceed that in the other 
vessel by an amount which may be regarded as the pressure due 
to the sugar molecules. This is the osmotic pressure of the 
solution. 

If the solution is weak, the molecules of solvent are in general 
far apart, and therefore interact only very weakly with one 
another (though they interact, of course, with the solvent mole- 
cules). In this respect the solute molecules in a weak solution 
may be said to behave similarly to the molecules of an ideal gas. 
This in turn leads to a number of analogies between the properties 
of weak solutions and those of ideal gases. 

We know that the pressure of an ideal gas is given by the 
formula p = NkT/V. It is found that the osmotic pressure p osm of 
a weak solution is given by an analogous formula, 

Posm = nkT/V, 

where V is the volume of the solution and n the number of 
molecules of solute in it. This is van't HofFs formula. 

It should be emphasised that the osmotic pressure of a weak 
solution (for a given volume and temperature) is determined only 
by the number of the solute particles and does not depend on 
their nature (or on the nature of the solvent), just as the pressure 
of an ideal gas is independent of its nature. As an example, we 
may mention that the osmotic pressure of a solution with a 
concentration of 1/10 mole/litre is 2-24atm. The osmotic 
pressure of sea water is about 2-7 atm. 

If we have a weak solution of several substances in the same 
solvent, then, from the above discussion, the osmotic pressure 
of the solution is determined by the total number of dissolved 
particles. It is therefore equal to the sum of the "partial" osmotic 
pressures of the individual solutes (corresponding to Dalton's 
law for gases). This should be borne in mind also when the 



238 SOLUTIONS [x 

dissolution is accompanied by decomposition of the molecules 
into parts (dissociation); this phenomenon will be discussed 
in §§89 and 90. The osmotic pressure of such a solution depends 
not only on the total quantity of solute but also on the degree 
to which its molecules dissociate. 

The analogy between a weak solution and an ideal gas extends 
further. For example, the height distribution of solute molecules 
in a field of gravity is given by a formula similar to the barometric 
formula (§54). This effect may be observed particularly clearly 
by using, instead of an ordinary solution, an emulsion consisting 
of very small particles of a substance suspended in a liquid. Since 
the mass of such particles is many times greater than that of the 
individual molecules, the variation of their concentration with 
height is seen from the barometric formula to be much more 
rapid, and is therefore easily observed directly. [In the baro- 
metric formula we must, of course, substitute the mass m of a 
particle in the emulsion minus the mass m of the liquid displaced 
by it, in accordance with Archimedes' principle.] 

§81. Raoult'slaw 

We know that, for a given pressure, there is a definite temper- 
ature, the boiling point, at which a liquid changes into a vapour. 
Let us now suppose that a non-volatile substance (i.e. one which 
does not vaporise when a solution of it in the liquid evaporates) 
is dissolved in the liquid; for example, sugar dissolved in water. 
It is found that the boiling point of the solution is different from 
that of the pure solvent (at the same pressure). 

From Le Chatelier's principle it is easy to deduce that the boil- 
ing point is raised when a solute is added. Let us consider a 
solution of sugar in water, in equilibrium with the vapour, and 
let a further quantity of sugar be added to the solution. The 
concentration of the solution is increased, and the system is no 
longer in equilibrium. Processes must occur in it which tend 
to oppose the external interaction, i.e. to decrease the concen- 
tration. For this to be so, the boiling point must rise, so that 
some of the vapour condenses into water. 

The rise in the boiling point of the solution is shown in the 
pT diagram by the fact that the evaporation curve of the solution 
(curve 2 in Fig. 101) is somewhat to the right of curve 1, the 
evaporation curve of the pure solvent. At the same time, as the 



§81] raoult's law 239 

diagram shows, curve 2 lies below curve 1. This means that 
the saturated vapour pressure of the solvent above the solution 
is less than that of the pure solvent at the same temperature. 
The decrease 8p in the saturated vapour pressure and the rise 
8T in the boiling point when the solute is present are shown in 
the diagram by the vertical and horizontal lines between the two 
curves. These quantities can be calculated if the solution is a 
weak one, as will be assumed below. 




Fig. 101. 

Let us return to the equilibrium shown in Fig. 100 between 
pure water and a solution, separated by a semipermeable parti- 
tion, and suppose that the whole system is in a closed space 
filled with saturated water vapour. Since the gas pressure in a 
field of gravity decreases with increasing height, the vapour 
pressure over the surface of the water will be greater than that 
over the solution, in accordance with the foregoing discussion. 
The pressure difference dp is clearly that due to a column of 
vapour of height h: 

S/> = Pvgh, 

where p v is the density of the vapour. The height h is determined 
by the osmotic pressure p osm of the solution: the pressure of the 
column of liquid balances the pressure p osm . The formula for the 
osmotic pressure gives 

Pigh = /?osm = nkT/Vt ; 

we shall take n to be here the number of molecules of solute 
per unit mass of liquid, so that Vi is the specific volume of the 
liquid, V t = 1/p,. Hence, substituting gh = nkT in the expression 
for dp, we obtain 

8p = p v nkT = nkT/V v . 



240 SOLUTIONS [x 

Finally, regarding the vapour as an ideal gas, we have its specific 
volume V v = NkT/p, where N is the number of molecules per 
unit mass of vapour or, what is the same thing, per unit mass of 
water. The final result is 

dplp = n/N. 

This is the required formula: the relative decrease in the vapour 
pressure is equal to the molecular concentration of the solution, 
i.e. the ratio of the numbers of solute and solvent molecules 
(or, what is the same thing, the ratio of the numbers of gram- 
molecules). This is called Raoult's law. We see that the change 
in the vapour pressure over the solution is independent of the 
specific properties of the solvent and solute; only the numbers of 
molecules are involved. 

This latter property does not hold good for the other quantity 
under discussion, the rise 8T in the boiling point. This is easily 
found by noting that the small quantities 8p and 8T are related by 

8p = {dpldT)8T. 
Using the Clausius-Clapeyron equation 

dp/dT = qp/RT 2 
(where q is the molar heat of evaporation; see §68), we obtain 

8T = (RT 2 lqp)8p. 
Finally, substituting 8plp = n/N, we find 

8T = RT 2 nlqN. 

The presence of a solute also affects the freezing point of a 
liquid. In the great majority of cases the solute does not enter 
the solid phase, i.e. pure solvent freezes out of the solution. Just 
as for evaporation, we can use Le Chatelier's principle and easily 
prove that the presence of solute lowers the freezing point. It is 
also found that the quantitative formula for the amount of this 
depression 8T is the same as the formula derived above for the 



§82] 



BOILING OF A MIXTURE OF LIQUIDS 



241 



change in the boiling point, q now signifying the molar heat of 
fusion of the solvent. 

The lowering of the freezing point is often used to determine the 
the molecular weight (the cryoscopic method). After dissolving a 
known weight of the substance under investigation, we determine 
8T, and hence calculate from the above formula the number of 
dissolved molecules, and so the molecular weight. The molecular 
weight can similarly be determined from the rise in the boiling 
point. 

§82. Boiling of a mixture of liquids 

When a mixture of two liquids boils, both components of the 
mixture generally vaporise, so that we have an equilibrium of a 
liquid and a gaseous phase each of which is a mixture. The 
resulting phenomena can be most clearly represented by means 
of a phase diagram, with the concentration c of the mixture on 
one axis and the temperature T or the pressure p on the other. 
Here we shall consider cT diagrams for a given value of the 
pressure. 

There are various types of phase diagram for the boiling of a 
liquid mixture. Here we shall consider those which occur for 
substances that mix in any proportions in the liquid state. 




100 %N, 



100% 0, 



Fig. 102. 



As an example of the first type we take a mixture of liquid 
oxygen and liquid nitrogen (Fig. 102). One of the vertical axes 
in the diagram corresponds to pure nitrogen and the other to 
pure oxygen, and between them lie all intermediate concentrations. 

The region above the upper curve corresponds to states of 
the high-temperature phase, i.e. the gaseous mixture, and the 



242 SOLUTIONS [x 

region below the lower curve corresponds to states of the liquid 
mixture. The hatched region between the two curves corresponds 
to equilibrium between liquid and vapour, the conditions of the 
liquid and vapour in equilibrium being determined by the points 
of intersection of the horizontal line through a given point with 
the two curves. For example, at the point a equilibrium exists 
between a gas whose composition is given by the abscissa of the 
point b and a liquid represented by the point c; the relative 
quantities of gas and liquid are inversely proportional to the 
lengths of ab and ac. The upper curve ADB is called the vapour 
curve and the lower curve ACB the liquid curve. A phase diagram 
of this shape is often referred to as a "cigar". 

The points A and B represent the boiling points of pure nitrogen 
and oxygen. Suppose that we have a liquid mixture whose com- 
position corresponds to the vertical line GH in Fig. 102. When 
such a mixture is heated, its state will vary along the line GC 
until the point C is reached. At this temperature the liquid begins 
to boil, but the composition of the vapour which boils off is not 
the same as that of the liquid: it is the composition which can be 
in equilibrium with the liquid at this temperature, i.e. that which 
is given by the point D. Thus the vapour which boils off has a 
higher nitrogen concentration than the liquid. Accordingly, the 
composition of the liquid will move towards an increasing content 
of oxygen. On further heating, therefore, the point representing 
the state of the liquid will move upwards along the curve CB. The 
vapour which boils off will be represented by a point which moves 
upwards along the curve DB. 

We see that the mixture does not boil at a constant temperature, 
unlike a pure liquid. The point at which boiling ceases depends 
on the conditions under which it occurs. If the vapour which 
boils off remains in contact with the liquid, the total composition 
of the liquid and vapour remains fixed, and the states of the 
system are always represented by points on the line GH. Hence 
we see that boiling begins at the point C and ends at the tem- 
perature of the point E where the vertical line GH intersects the 
upper curve of the "cigar". 

If, however, boiling takes place in an open vessel and the 
vapour which boils off is steadily removed, only the vapour 
which has just boiled is in equilibrium with the liquid at any 
instant. A quantity of liquid, on boiling, is converted entirely 



§82] BOILING OF A MIXTURE OF LIQUIDS 243 

into vapour, i.e. the resulting quantity of vapour must have the 
same composition as the boiling liquid. Thus the last part of the 
boiling process occurs at a point where the composition of the 
liquid and that of the vapour are the same, i.e. at the boiling point 
B of pure oxygen. 

Exactly similar effects occur in the condensation of a vapour 
into a liquid. 

Another type of phase diagram occurs, for example, for a 
mixture of chloroform and acetone (Fig. 103). This differs from 
the previous case in that the two curves have a maximum point 
A , at which they touch. Here again, the region between the curves 
corresponds to liquid and vapour in equilibrium, while the regions 
above and below the curves correspond to the gaseous and liquid 
phases. 



T 


Gas 




A(63°C) 


61 °C 


^^ ^^, 




""" \ 



Chloroform 



56° C 



Acetone 



Fig. 103. 



Boiling or condensation occurs in a similar manner to the 
preceding case. For example, when liquid boils in an open vessel, 
the points representing the states of the liquid and the vapour 
move upwards along the two curves, but the process now 
terminates not at the boiling point of one of the pure components 
but at the point A where the curves touch. At this point the 
composition of the liquid is the same as that of its vapour. A 
mixture whose composition corresponds to the point A (called 
an azeotropic mixture) therefore boils away completely at a 
constant temperature, as if it were a pure substance. 

Finally, there are mixtures (for example, carbon disulphide 
and acetone) whose phase diagrams differ from the preceding 
type only in that the curves have minima instead of maxima 
(Fig. 104). 

The effects described above are widely used in practice in 
order to separate the components of various mixtures. In its 



244 



SOLUTIONS 



[x 



46°C 




56 °C 



100% CS 2 



Acetone 



Fig. 104. 



simplest form, the method of fractional distillation consists 
in collecting and condensing the initial fractions of the vapour 
boiling off from a liquid mixture, and then redistilling the resulting 
substance. For example, when a mixture of alcohol and water 
boils, the vapour formed has a higher content of the more volatile 
alcohol than is present in the liquid. By condensing the initial 
fractions of this vapour and again boiling the resulting liquid, 
we can separate the water and the alcohol more and more 
completely. When the phase diagram is of the type shown in 
Fig. 102 the components of the mixture can, in principle, be 
completely separated by repeating the process several times. 
When the phase diagram is as in Fig. 103 or 104, however, 
complete separation is not possible: only an azeotropic mixture 
can be separated, together with one or the other pure substance, 
depending on the composition of the original mixture. The 
mixture of water and alcohol mentioned above is of this type; it 
has a minimum boiling point at a composition of 95-6% by weight 
of alcohol. The alcohol cannot be further purified by fractional 
distillation. 



§83. Reverse condensation 

The existence of critical points for liquid-gas transitions in 
pure substances has the result that critical phenomena occur 
in mixtures also. Without analysing all possible variations, let 
us consider some characteristic features of these phenomena. 

The phase diagram of oxygen-nitrogen mixtures shown in 
Fig. 102 refers to a pressure of 1 atm. At higher pressures the 
diagram remains of the same type, but only up to the critical 
pressure of one of the pure components, in this case nitrogen, 
at 33-5 atm (the critical pressure of oxygen is 49-7 atm). Since 



§83] 



REVERSE CONDENSATION 



245 



pure nitrogen cannot separate into phases above this point, it is 
evident that the "cigar" in the phase diagram of the mixtures 
must become "detached" from the vertical axis and become of 
the form shown in Fig. 105. We see that a point K is now present 
in the diagram at which the two coexisting phases become 
identical; this is called the critical point. Here again the presence 
of the critical point means that a continuous transition is possible 
between liquid and gas, so that the distinction between these two 
phases becomes arbitrary. 




Fig. 105. 

The condensation of a gas mixture may be accompanied by 
unusual phenomena when there is a critical point in the phase 
diagram. We may illustrate these by means of a cp diagram (for 
a given value of the temperature), which more nearly corresponds 
to the usual conditions under which they are actually observed. 

Figure 106 shows part of such a phase diagram near the critical 
point K; unlike the cT diagrams used above, the gas phase cor- 
responds to the region below the hatched area, i.e. the region of 
low pressures. 




Composition 
Fig. 106. 



246 



SOLUTIONS 



[x 



Let us consider a mixture whose composition corresponds to 
the vertical line AC. In isothermal compression, when the point 
B is reached, the mixture begins to condense, forming a liquid 
phase B'. As the pressure is further increased, the quantity of 
liquid at first increases but later decreases, and when the point 
C is reached the liquid (which is then represented by the point 
C) disappears entirely. This phenomenon is called reverse 
condensation. 

§84. Solidification of a mixture of liquids 

The phase diagram for a liquid and a solid can be represented 
in the same way as that for a liquid and a gas. We again plot the 
concentration of the mixture (percent by atoms) as abscissa, and 
the temperature as ordinate, and consider the diagram for a given 
pressure. 



960 °C 




Fig. 107. 



If the two substances mix in any proportions both in the liquid 
and in the solid state, then the form of the diagrams is exactly 
similar to the liquid-gas phase diagrams discussed in §82. For 
example, an alloy of silver and gold has the phase diagram shown 
in Fig. 107. The region above the curves corresponds to liquid 
mixtures of the two metals, and the region below the curves 
corresponds to solid alloys. The process of melting of the alloy 
occurs similarly to the boiling of a liquid mixture as described 
in connection with Fig. 102. 

The phase diagram for the bismuth-cadmium system, shown 
in Fig. 108, is of an entirely different type. This system has the 
property that the two components form no solid solutions. 

The region which is not hatched corresponds to liquid mix- 
tures. In all the other regions there is separation into two phases. 
In region I the two phases are solid crystals of pure cadmium 
(represented by the left-hand vertical axis) and a liquid (re- 



§84] SOLIDIFICATION OF A MIXTURE OF LIQUIDS 247 

presented by the curve AO). For example, at a point d in this 
region there is equilibrium between phases represented by the 
points where the horizontal line ef meets the ordinate axis (pure 
cadmium) and the curve AO (liquid mixture); the quantities of 
these phases are inversely proportional to the lengths de and df. 
Similarly, in region II the solid phase is bismuth in equilibrium 
with a liquid whose composition is determined by the curve OB. 
Finally, in region III there is a mixture of solid crystals of 
cadmium and bismuth. 



321 °C 




271 °C 



144 °C 



Fig. 108. 



The points A and B are the melting points of pure cadmium 
and pure bismuth. The curve AOB gives the temperature at which 
liquid mixtures of the two components begin to solidify. 

Let us consider, for instance, the process of solidification of a 
liquid mixture whose composition is given by the vertical line 
ab. The solidification begins at the temperature of the point b at 
which this vertical line intersects the curve A O, and crystals of 
cadmium separate from the liquid. As the temperature decreases 
further, the liquid mixture becomes richer in bismuth, and the 
point representing it moves downwards along the curve bO until 
the point O is reached. The temperature then remains constant 
until the liquid has completely solidified. At the temperature of 
O, crystals of the remaining cadmium and all the bismuth that 
was in the liquid are formed. 

The point O is called the eutectic point. It is a point at which 
three phases are in equilibrium: solid cadmium, solid bismuth, 
and the liquid mixture. The crystalline mixture which solidifies 
out at the eutectic point consists of very small crystals of each 
component, called a eutectic mixture. To the right of O, in region 
III, the eutectic mixture contains larger crystals of previously 
solidified bismuth, and to the left, crystals of cadmium. 



248 



SOLUTIONS 



[x 



Figure 109 shows the typical form of the "cooling curve" 
corresponding to Fig. 108. Here the temperature of the system 
is plotted as a function of time during slow cooling of a liquid 
of given composition (in this case corresponding to the vertical 
line ab). When the point b is reached, a break occurs on the 
cooling curve; because solidification begins, which is accompanied 
by the evolution of heat, the cooling becomes somewhat slower. 
At the temperature of the eutectic point there is a "plateau": 
the curve has a horizontal section, corresponding to the complete 
solidification of the alloy at a constant temperature. The recording 
of such cooling curves is the basis of the derivation of phase 
diagrams by the method of thermal analysis. 



Time 
Fig. 109. 

The phase diagram of the silver-copper system, shown in 
Fig. 110, differs from the previous one in that each of its solid 
components can dissolve a certain amount of the other. The 
diagram therefore contains three single-phase regions: in addition 
to region I (liquid mixtures), we have region II (solid solutions of 
copper in silver) and region III (solid solutions of silver in 
copper). 




779 °C 



Fig. 110. 



§84] 



SOLIDIFICATION OF A MIXTURE OF LIQUIDS 



249 



Finally, let us consider the phase diagram of the aluminium- 
calcium system (Fig. 111). In this case, although the two com- 
ponents do not form solid solutions, there exist certain chemical 
compounds; we might say that only solid solutions of certain 
definite compositions exist. The vertical line BD corresponds to 
the compound CaAl 2 . The point B is the melting point of this 
compound, and is the maximum of the curve ABC. Another 
compound, CaAl4, decomposes before melting begins. The 
vertical line EF corresponding to this compound therefore does 
not reach the boundary AB of the liquid state. All the hatched 
areas are regions of separation into two phases. The two phases 
that are in equilibrium are always given by the points of inter- 
section of a horizontal line with the two nearest vertical lines in 
the diagram. For example, in region I the liquid is in equilibrium 
with crystals of the compound CaAl 2 ; in region II, with crystals 
of CaAl 4 ; in region III, crystals of aluminium with crystals of 
CaAl 4 , and so on. 




We may note that the study of phase diagrams by means of 
thermal analysis can itself provide information as to the existence 
of solid chemical compounds of various substances. The exis- 
tence of a compound is shown by the presence on the melting 
curve of a maximum (as at B in Fig. Ill) or a break (as at A). 

There is a great variety of phase diagrams for different mix- 
tures. The few diagrams described here are among the simplest. 
These examples show, however, the characteristic properties 
and types which may be seen in more complicated diagrams also. 



250 SOLUTIONS [x 

§85. The phase rule 

We shall now recapitulate and generalise some of the properties 
of phase equilibrium described in Chapters IX and X. 

The thermal state of a homogeneous body consisting of a 
single substance is defined by the values of two independent 
quantities, the temperature T and the pressure p. If a further 
phase of the same substance is added (e.g. ice and water), these 
phases can coexist not at all values of p and T but only when a 
certain relation holds (represented by a curve in the pT diagram). 
We may say that the equilibrium with ice imposes on the equation 
of state of water a further condition, as a result of which the 
number of independent quantities is reduced from two (p and T) 
to one (p or T). 

Three phases of one substance (e.g. water, ice and steam) can 
coexist only for certain definite values of p and T, where the 
ice-water and steam-water equilibrium curves intersect. We 
may say that the addition of a third phase imposes a further 
condition, as a result of which the number of independent 
quantities is reduced to zero. 

Hence it is clear that four phases of one substance (e.g. water, 
steam, and two forms of ice) can not exist in equilibrium. Such 
an equilibrium would require three added conditions to be satis- 
fied, and this can not be achieved by means of the two disposable 
variables p and T. 

Let us now consider a body consisting of two substances, such 
as a liquid solution. Its state is defined by three independent 
variables: the temperature T, the pressure p and the concentra- 
tion c. Let this solution be in equilibrium with its vapour (con- 
taining the same two substances). This imposes a certain further 
condition, and only two of the three quantities describing the 
state of the solution remain arbitrary. Thus equilibrium between 
a liquid solution and a vapour is possible at any pressure and 
temperature (for example), but the concentration of the solution, 
and therefore that of the vapour, must then have a definite value. 
We have already seen this from the phase diagrams in the present 
chapter. 

If a further phase is added which consists of the same two 
substances, this imposes a further condition, and only one 
variable remains arbitrary. For instance, at a given pressure 
three phases can coexist at only one point, with a definite tern- 



§85] THE PHASE RULE 251 

perature and definite concentrations. The eutectic point in the 
phase diagrams in §84 is a point of this type. 

Finally, four phases of two components can be in equilibrium 
only for certain values of all the quantities (pressure, temperature 
and concentrations), and the equilibrium of five (or more) phases 
is impossible. 

These statements are easily generalised to the equilibrium of 
phases containing any number of components. Let the number 
of components be n, and the number of coexisting phases r; 
and let us consider one phase. Its composition is specified by 
the values of n— 1 concentrations, for example the ratios of the 
quantities of each of n — 1 components to that of the nth. Thus 
the state of the phase is defined by n+l quantities: p, T and 
n—\ concentrations. But this phase is in equilibrium with r— 1 
other phases, which imposes r — 1 extra conditions on its equation 
of state. These must not exceed the number of variables, i.e. 
n+l must not be less than r — 1 , or in other words 

r «£ n + 2. 

Thus not more than n + 2 phases composed of n substances can 
coexist in equilibrium. This is called the phase rule. 

When the maximum number (n + 2) of phases coexist, all the 
quantities which describe the states of the phases (p, T and the 
concentrations of all the phases) must have definite values. When 
r phases are in equilibrium, the values of (n + 1) — (r — 1) = n + 2 — r 
quantities may be specified arbitrarily. 



CHAPTER XI 

CHEMICAL REACTIONS 



§86. Heats of reaction 

This chapter will deal with chemical reactions from the 
physical point of view, having regard to properties which are 
common to all reactions, whatever the chemical nature of the 
reacting substances. 

Any chemical reaction is accompanied by absorption or 
evolution of heat. In the former case the reaction is said to be 
endothermic, and in the latter case exothermic. It is clear that, 
if a reaction is exothermic, the reverse reaction is endothermic, 
and vice versa. 

The heat involved in a reaction depends in general on the condi- 
tions under which it occurs. Hence, strictly speaking, we should 
distinguish the quantity of heat according as the reaction occurs 
at constant pressure or at constant volume. In practice, however, 
the difference is usually very slight. 

The heat of reaction is written in the reaction equation with a 
positive sign on the side where heat is evolved, or with a negative 
sign where it is absorbed. For example, the equation 

C + O 2 = CO 2 + 400kJ 

signifies that 400 kJ of heat are evolved in the combustion of 
one gram-atom of carbon (graphite). Two further examples are 

iH 2 + iCl 2 = HCl + 92, 

iN 2 + |H 2 =NH 3 + 46; 

here, as in all subsequent examples, the heat is again stated in 
kilojoules per mole of the reacting substances. 

In the above examples it has been assumed that all the sub- 
stances (except graphite) are in the gaseous state (at room 

252 



§86] HEATS OF REACTION 253 

temperature and atmospheric pressure). The aggregate state of 
the reacting substances must be specified, since the heat of 
reaction depends on this state, and the dependence may be very 
considerable. As an example, let us find the difference between 
the heats of formation of liquid water and steam from gaseous 
oxygen and hydrogen. The heat of evaporation of a gram- 
molecule of water (at 20°C) is 44 kJ, i.e. 

H 2 0(gas) = H 2 0(liq) + 44. 
Adding this to the equation of formation of steam, 

H 2 + K) 2 = H 2 0(gas) + 240, 
we obtain the equation of formation of liquid water, 

H 2 + K) 2 = H 2 0(liq) + 284. 

The heat of reaction also depends, of course, on the tempera- 
ture at which the reaction takes place. The value is easily con- 
verted from one temperature to another if the specific heats of 
all the reacting substances are known, in the same way as we 
have just converted the value from one aggregate state to another. 
To make the conversion it is necessary to calculate the heat 
required to bring all the substances which participate in the 
reaction from one temperature to another. 

If several reactions occur in succession, it follows from the 
law of conservation of energy that the total heat of the whole 
sequence of reactions is equal to the sum of the heats of each 
successive reaction. We can say, moreover, that, if we start 
from certain given substances and produce other substances by 
a series of intermediate reactions, the total heat evolved is 
independent of the intermediate stages through which the reaction 
took place. 

By means of this rule we can, in particular, calculate the heats 
of reactions which in practice could never occur. Let us find, 
for example, the heat of formation of acetylene gas directly from 
the elements carbon (graphite) and hydrogen: 2C + H 2 = C 2 H 2 . 
This reaction cannot occur directly and in practice is brought 
about by other means; the heat of reaction therefore cannot be 



254 CHEMICAL REACTIONS [XI 

measured directly. We can, however, calculate this heat from 
the known (directly measurable) heats of combustion of carbon, 
hydrogen and acetylene itself: 

2C + 2O 2 =2CO 2 + 800, 

H 2 + K) 2 = H 2 O + 240, 

C 2 H 2 + f 2 = 2C0 2 + H 2 + 1 300. 

Adding the first two equations and subtracting the third, we 
obtain 

2C+H 2 = C 2 H 2 -260. 

The heat of formation of a compound from its elements 
depends on their state. In physics the heat of formation from 
the atoms of the elements is of interest, not that from the elements 
in their natural state. The heat of formation from the atoms 
determines the internal energy of the compound as such and is 
independent of the state of the initial substances. Some examples 
are 

2H = H 2 + 435, 
2O = O 2 + 500, 

C(atoms) = C(graphite) + 720, 
2C(atoms) + 2H = C 2 H 2 + 1600. 

The heat of formation of a compound from the elements may 
be either positive or negative, but the heat of formation from 
the atoms is always positive, since otherwise the compound would 
be unstable and could not exist. 

§87. Chemical equilibrium 

As a chemical reaction proceeds, the quantities of the original 
substances decrease, and reaction products accumulate. Ulti- 
mately, the reaction leads to a state in which the quantities of all 
the substances no longer vary. This is called a state of chemical 
equilibrium, a particular case of thermal equilibrium. 



§87] CHEMICAL EQUILIBRIUM 255 

In chemical equilibrium there is generally present a certain 
quantity of the original substances as well as the products 
formed in the reaction. It is true that in many cases this quantity 
is very small, but this does not, of course, affect the principle. 

The establishment of chemical equilibrium in which both 
initial and final substances are present occurs for the following 
reason. Let us consider, for example, a reaction between hydrogen 
and iodine gases to form hydrogen iodide: 

H 2 + I 2 = 2HI. 

As well as the formation of HI from H 2 and I 2 , in a mixture of 
these three substances the reverse process of dissociation of 
HI into hydrogen and iodine will also necessarily occur: the 
forward reaction is always accompanied by the reverse reaction. 
As the quantity of HI increases and that of H 2 and I 2 decreases, 
the forward reaction will obviously become slower and the 
reverse reaction quicker, and a point is finally reached at which 
the rates of the two reactions become equal, with the same 
number of new HI molecules formed as dissociate in the same 
time. The quantities of all the substances thereafter remain 
unchanged. 

Thus chemical equilibrium (and in fact other types of thermal 
equilibrium) is dynamic on the molecular scale; the reactions do 
not actually cease, but the forward and reverse reactions occur at 
equal rates and therefore produce no overall effect. 

It is clear that, if the reaction in the above example begins 
from a mixture of hydrogen and iodine, the relative quantities of 
all three substances in the equilibrium state will be the same as in 
a reaction which begins with the decomposition of pure HI. The 
chemical equilibrium position does not depend on the side from 
which it is approached. 

Moreover, the chemical equilibrium also does not depend on 
the conditions under which the reaction occurs or on the inter- 
mediate stages through which it passes. The position of equilib- 
rium depends only on the state of the substances in equilibrium, 
i.e. on the temperature and pressure of the equilibrium mixture. 

When the temperature changes, the position of chemical equilib- 
rium is altered. The direction of this shift depends entirely on the 
heat of reaction, as is easily seen by means of Le Chatelier's 



256 CHEMICAL REACTIONS [XI 

principle. Let us consider an exothermic reaction, such as the 
formation of ammonia from nitrogen and hydrogen (N 2 + 3H 2 
= 2NH 3 ), and assume that the reaction has already reached a 
state of equilibrium. If the equilibrium mixture is now heated, pro- 
cesses must occur in it which tend to cool it: a certain quantity 
of ammonia must decompose, and heat is thus absorbed. This 
means that the chemical equilibrium is shifted in the direction 
such that the quantity of ammonia is decreased. 

Thus the "yield" of exothermic reactions decreases when the 
temperature is raised; in endothermic reactions, on the other 
hand, the yield increases with increasing temperature. 

Similarly, the dependence of the equilibrium position on the 
pressure is related to the change in volume accompanying the 
reaction (at constant pressure). Increasing the pressure lowers 
the yield of reactions in which the volume of the reacting mixture 
increases, and raises that of reactions in which the volume de- 
creases. The latter case occurs, for instance, in the reaction of 
formation of gaseous ammonia: since the number of NH 3 mole- 
cules formed is less than the number of reacting N 2 and H 2 
molecules, the volume of the gas mixture decreases in the 
reaction. 

§88. The law of mass action 

We shall now give a quantitative formulation of the concept 
of chemical equilibrium. Let us first consider chemical reactions 
in a gas mixture, all the substances participating in a reaction 
being in the gaseous state. 

As an example we shall again take the reaction of formation 
of HI. The reaction between hydrogen and iodine can occur 
when H 2 and I 2 molecules collide. The rate of the reaction of HI 
formation (i.e. the number of HI molecules formed per unit 
time) is therefore proportional to the number of such collisions. 
This number in turn is proportional to the densities of hydrogen 
and iodine in the mixture, i.e. the numbers of molecules of 
hydrogen and iodine per unit volume. The density of a gas is 
proportional to its pressure. Thus the rate of the reaction of HI 
formation is proportional to the partial pressures of these gases 
in the mixture, i.e. is kip H2 pi 2 , where the coefficient k x depends 
only on the temperature. Similarly, the rate of the reaction of 
HI decomposition is proportional to the number of collisions 



§88] THE LAW OF MASS ACTION 257 

between HI molecules, and therefore to the square of the partial 
pressure of HI in the mixture; let it be k^Pm 2 - 

In equilibrium, the rates of the forward and reverse reactions 
are equal: 

kiPnJ>i2 = k&m 2 • 
Putting fc 2 /&i = K{T), we thus have 

PhiPiJPhi 2 = K(T). 

This equation relates the partial pressures of all three gases in 
the equilibrium state. The quantity K(T) is called the equilibrium 
constant for the reaction concerned. It is independent of the 
quantities of the reacting substances. The relation expressed by 
the above formula is called the law of mass action. 

This law can be written in a similar form for any other reaction 
between gases. It may be written in a general form as follows. 

In the chemical equation for the reaction we can arbitrarily 
transfer all the terms to the same side, writing e.g. 

H 2 + I 2 -2HI = 0. 

Any reaction may be represented in the general form 
^Ai+i^AaH = 0, 

where A 1} A 2 , . . . are the chemical symbols of the reacting sub- 
stances, and v u v 2 , . . . are positive or negative integers; for 
instance, in the above example v m = v l2 = 1 , v m = —2. Then the 
law of mass action takes the form 

Pi n Pf*-- = K{T), 

where p x , p 2 , . . . are the partial pressures of the various gases. 

It is often more convenient to use the concentration of the 
substances in the mixture, instead of the partial pressures. We 
define the concentration of the ith substance in the mixture as 
the ratio c t = NJN of the number N { of molecules of this sub- 
stance to the total number N of molecules in the mixture, or, 



258 CHEMICAL REACTIONS [XI 

what is the same thing, the ratio of the corresponding numbers 
of moles. Since the total pressure of the gas mixture is p = NkTjV, 
where V is the volume of the mixture, and the partial pressures 
Pi = NikTlV, we have 

Pi = CiP. 

Substituting these expressions in the equation of the law of 
mass action, we obtain the latter in the form 

c 1 n c 2 n •• • = K(T)p~^ +V2+ -) 

which relates the equilibrium concentrations of all the substances. 
The quantity on the right of this equation is also called the 
equilibrium constant, but may depend on the pressure as well 
as on the temperature. It is independent of the pressure only 
if the sum v x + v 2 + • • • = 0, i.e. if the total number of molecules 
is unchanged in the reaction (as, for example, in the reaction 
H 2 + I 2 = 2HI). 
For the reaction of ammonia formation, for instance: 

N 2 + 3H 2 = 2NH 3 , 

we have 

Cn 2 Ch 2 3 /cnh 3 2 = K(T)lp 2 - 

When the pressure increases, the right-hand side of this equation 
decreases, and the left-hand side must therefore decrease also. 
Thus the equilibrium concentrations of the initial substances 
decrease and that of ammonia increases, in agreement with the 
result previously found by means of Le Chatelier's principle. 
We have also seen that the yield of this reaction must decrease 
with increasing temperature; we can now say that in this case the 
equilibrium constant K(T) increases with temperature. 

The following comment must be made regarding the foregoing 
derivation of the law of mass action. In our discussion it has 
been assumed that the course of the reaction is as represented 
by the chemical equation. In the reaction of HI formation this 
is in fact so, but the majority of reactions do not really take 



§88] THE LAW OF MASS ACTION 259 

place in the way that would be expected from the equation; for 
example, the formation of an ammonia molecule does not occur 
by the collision of a nitrogen molecule with three hydrogen 
molecules. The representation of the reaction by a single equation 
is usually a mere summary of a sequence of intermediate steps, 
taking into account only the initial and final substances. We 
shall discuss this further below. The properties of chemical 
equilibrium and the law of mass action describing them are, 
however, independent of the actual reaction mechanism. 

To illustrate the application of the law of mass action, let us 
make a complete analysis of the simple reaction of dissociation 
of hydrogen: 

H 2 = 2H, 

and determine the degree of dissociation which is reached at 
equilibrium. Let the total number of hydrogen atoms (both 
isolated and in molecules) be A. The degree of dissociation x 
may be defined as the ratio of the number A/ H of hydrogen atoms 
to the total number of atoms A . Then 

N u =Ax, N H2 = 1>A(\-x), N = N n + N U2 = U(l+x). 

Expressing the concentrations c H and c H2 in terms of these 
quantities and substituting in the law of mass action, we find 

c H Jc H 2 = (l-x 2 )l4x 2 = pK, 

whence 

jc= llV{l+4pK); 

this determines, in particular, the way in which the degree of 
dissociation depends on the pressure. 

If several different reactions can occur in a gas mixture, the 
law of mass action must be applied to each reaction separately. 
For example, in a mixture of the gases H 2 , 2 , CO, C0 2 and 
H 2 0, the following reactions can occur: 

2H 2 = 2H 2 + 2 , 2CO + 2 = 2C0 2 . 



260 CHEMICAL REACTIONS [XI 

For these reactions we have 

Ph 2 o IPhz P02 = Ku 
Pco PoJPco 2 ~ K<l, 

and the state of chemical equilibrium is determined by the 
simultaneous solution of these two equations. In the mixture 
considered, other reactions can occur, for example 

H 2 + CO = C0 2 + H 2 

but this reaction need not be considered, as it is just the sum of 
the two reactions written above, and the law of mass action would 
give an equation which is simply the product of the two equations 
previously derived. 

Let us now consider a reaction which involves not only gases 
but also a solid. The reaction between the solid and a gas molecule 
can occur when the latter collides with the surface of the solid. 
The number of collisions of gas molecules with unit area of the 
surface obviously depends only on the density of the gas and not 
on the quantity of the solid. Accordingly, the rate of reaction 
per unit area of the surface of the solid will be proportional only 
to the partial pressure of the gases and does not depend on the 
quantity of the solid. Hence it is clear that the law of mass action 
is valid also for reactions which involve solids, with the differ- 
ence that the equation for it includes only the concentrations of 
the gases and not the quantity of solids. The properties of the 
latter affect only the temperature dependence of the equilibrium 
constant. 

For example, in the decomposition of limestone with the 
evolution of carbon dioxide, 

CaC0 3 = CaO + C0 2 , 

the only gas is C0 2 (since the calcium oxide remains solid). The 
law of mass action therefore gives simply 

p C02 = K(T). 



§88] THE LAW OF MASS ACTION 261 

This means that in equilibrium (at a given temperature) carbon 
dioxide over limestone must have a definite partial pressure. 
This is similar to the case of evaporation, where again the gas 
pressure over a body is determined only by the temperature 
and does not depend, for example, on the quantity of either 
substance. 

The law of mass action also holds good for reactions between 
substances in solution if the solution is a weak one; here again 
we have an analogy between the properties of gases and those 
of weak solutions, as already noted in §80. The derivation of 
the law of mass action for gas reactions has been based on a 
calculation of the number of collisions between molecules. A 
similar calculation can be made for reactions in solution; the 
fact that the reacting molecules are not in a vacuum but in a 
medium, the solvent, affects only the dependence of the equilib- 
rium constant on temperature and pressure. In the equation of 
the law of mass action, 

c 1 v *c 2 t *--- = K(p,T), 

the dependence of K on both temperature and pressure therefore 
remains unknown. The concentrations c 1 ,c 2 ,... in this equation 
are now defined as the quantities of solutes in a given quantity 
(or per unit volume) of the solvent. 

A similar formula is valid for reactions which involve not only 
the solutes but also the solvent, for example the hydrolysis of 
cane sugar to glucose and fructose which occurs in an aqueous 
sugar solution. Since the number of water molecules is much 
greater than the number of sugar molecules (the solution being 
assumed weak), the concentration of water is practically un- 
changed by the reaction. Thus, in the equation of the law of mass 
action, only the solute concentrations need be included: 

[cane sugar] = R , T , 
[glucose] [fructose] 

where the square brackets denote molar concentrations, i.e. 
numbers of moles per litre of water. 



262 CHEMICAL REACTIONS [XI 

§89. Strong electrolytes 

A large number of substances are present in solution not as 
molecules but as charged constituents of molecules called ions; 
the positive ions are known as cations and the negative ions as 
anions. Such substances are said to be strong electrolytes. A 
substance which dissolves in the form of ions is said to dissociate 
in solution, and this phenomenon is called electrolytic dissociation. 

In solution in water, almost all salts are strong electrolytes, and 
so are some acids (such as HC1, HBr, HI and HN0 3 ) and some 
bases (such as NaOH and KOH). In salts the cation is the metal 
and the anion is the acid radical (e.g. NaCl — » Na + + CI - ). Acids 
dissociate to form the cation H + and the acid-radical anion 
(HN0 3 — »H + + N0 3 - ); alkalis give the metal cation and the 
anion OH", called hydroxyl (NaOH -» Na + + OH"). 

The phenomenon of electrolytic dissociation is observed in 
some other solvents also, but it appears most strongly in aqueous 
solutions. 

If two strong electrolytes are simultaneously dissolved in 
water, for instance NaCl and KBr, there is no reason to regard 
the solution as one of NaCl and KBr specifically; it contains only 
the separate ions K + , Na + , CI - and Br - , and the same solution 
could just as well (or rather, with just as little meaning) be called 
a solution of NaBr and KC1. 

In reactions between strong electrolytes in solution, only the 
separate ions actually take part, since there are no undissociated 
molecules in the solution. The heat of reaction between strong 
electrolytes therefore depends only on the ions which directly 
participate in the reaction, and not on which other ions are present 
in the solution (if the solution is weak, of course). Let us consider, 
for example, the neutralisation of strong acids with alkali. It is 
inaccurate to write the neutralisation reaction as, for example, 
NaOH + HC1 = NaCl + H 2 0; in reality, only the H + and OH~ 
ions react, combining to form water: H + + OH~ = H 2 0. This 
reaction is obviously the same for all strong acids and alkalis, 
whatever the metal and the acid radical. The heat of reaction is 
therefore likewise the same for the neutralisation of any strong 
acid by any strong alkali. The value of this heat for one mole of 
acid and one mole of alkali is 57 kJ: 

H + + OH- = H 2 0+57kJ. 



§89] STRONG ELECTROLYTES 263 

Let us next consider a saturated solution of any strong elec- 
trolyte whose solubility is low, for example silver chloride in 
water. By the definition of saturation this solution is in equilib- 
rium with solid silver chloride. This equilibrium may be regarded 
as the chemical equilibrium of the reaction 

Ag + + Cl- = AgCl, 

where Ag + and CI - are in the solution and AgCl in the solid 
state: the number of AgCl molecules which go into solution per 
unit time is equal to the number of molecules deposited from the 
solution by combination of ions per unit time. Since, on the 
other hand, the solution is weak (because the solubility of silver 
chloride is low), we can apply the law of mass action. Here, as 
explained previously, only the concentrations of the solutes need 
be included, and we find 

[Ag + ][C1-] = K, 

where the square brackets again denote molar concentrations 
(numbers of moles per litre of water). The constant K (which is, 
of course, a function of temperature) is called the solubility 
product for the electrolyte concerned. For example, for silver 
chloride at room temperature ^=lx 10 -10 (mole/1) 2 ; for CaC0 3 , 
K = 1 X 10- 8 (mole/1) 2 . 

Thus the product of the concentrations of anions and cations 
in a saturated solution of a strong electrolyte of low solubility is 
a constant. If no salt containing silver or chlorine ions, except 
silver chloride itself, is present in solution in water, the con- 
centrations [Ag + ] and [CI - ] are equal to the solubility c of silver 
chloride. Hence it follows that 

K = c 2 . 

Now let a quantity of another chloride, of high solubility (such 
as NaCl), be added to a saturated solution of AgCl. Then some 
of the latter will be deposited as solid from the solution, since the 
addition of NaCl raises the concentration of chloride ions while 
that of silver ions remains unchanged; some of the AgCl must 
therefore be deposited in order that (he product [Ag + ][Q~] may 
remain constant. 



264 CHEMICAL REACTIONS [XI 

§90. Weak electrolytes 

As well as strong electrolytes, there are substances which 
dissociate in solution but do so only partly; in solutions of these 
substances there are not only ions but also neutral molecules. 
Such substances are called weak electrolytes. The majority of 
acids and bases, and some salts (such as HgCl 2 ), are weak 
electrolytes in aqueous solution. 

The law of mass action is applicable to weak solutions of weak 
electrolytes. Let us consider, for example, a solution of acetic 
acid (CH3COOH), which dissociates in water according to the 
equation 

HAc = H + + Ac~ 

(the symbol Ac denoting the acid radical CH 3 COO). Dissocia- 
tion continues until equilibrium is established, when the ion 
concentrations satisfy the equation 

[Ac-][H + ]/[HAc] = K. 

The constant K is called the dissociation constant. For instance, 
for acetic acid at room temperature K = 2X 10 -5 mole/litre. 

A dissociation reaction is endothermic, i.e. it occurs with 
absorption of heat. As with all endothermic reactions, its "yield" 
increases with rising temperature, i.e. the dissociation constant 
increases. 

The dissociation constant is independent of the quantity of 
dissolved electrolyte (so long as the solution remains weak) and 
is a fundamental property of the electrolyte, but the degree of 
dissociation (i.e. the ratio of the number of dissociated molecules 
to the total number of electrolyte molecules) depends on the 
concentration of the solution. 

Let a total of c moles of electrolyte be dissolved in a litre of 
water, and let the degree of dissociation be a. Then the number 
of dissociated moles is ca. If an electrolyte molecule dissociates 
into one anion and one cation (as in the example of acetic acid 
considered above), then the concentration of each is ca. The 
concentration of undissociated molecules is c(l — a). The law of 
mass action therefore gives 

a 2 cl(\-a) = K. 



§90] WEAK ELECTROLYTES 265 

Hence we find the degree of dissociation in terms of the 
concentration of the solution: 

_ -K + y/(K 2 + 4Kc) = IK 

a 2c K + V(K 2 + 4Kcy 

This formula shows that, as the concentration c decreases, the 
degree of dissociation increases, tending to unity at infinite dilu- 
tion (i.e. as c — > 0). Thus, the more dilute the solution, the more 
the electrolyte is dissociated. This naturally follows from the 
fact that a molecule dissociates under the action of water mole- 
cules, which are present everywhere, but for recombination to 
occur two different ions must come together, and this occurs more 
rarely in more dilute solutions. 

Water is itself a very weak electrolyte. A very small fraction of 
its molecules are dissociated in accordance with the equation 

H 2 = H + + OH- 

Since H 2 is at the same time the solvent with respect to the 
ions H + and OH~, the formula for the law of mass action need 
include, as we know, only the concentrations of these ions: 

[H + ][OH-] = K. 

For pure water at 25°C, 

K = 10~ 14 (mole/litre) 2 . 

Since in pure water the concentrations of H + and OH~ ions are 
evidently equal, we find that each is 10~ 7 . Thus one litre of water 
contains only 10 -7 mole of H + ions (and the same quantity of 
OH~); 1 mole of water (18 g) is dissociated only in ten million 
litres. 

The decimal logarithm of the concentration of H + ions, with 
sign reversed, is called the pH: 

pH = -log 10 [H + ]. 

For pure water at 25°C the pH is 7-0; at 0°C it is 7-5 and at 60°C 
6-5. 



266 CHEMICAL REACTIONS [XI 

When acids dissolve they release H + ions. But the product of 
concentrations [H + ][OH~] must remain constant and equal to 
10~ 14 . Some of the OH~ ions must therefore combine with H + 
ions to form neutral molecules of water. Thus the concentration 
[H + ] is greater than its value in pure water (10~ 7 ), and the pH 
of an acid solution is consequently less than 7. Similarly, in 
solutions of alkalis (which release OH - ions) the pH is greater 
than 7. The pH of a solution is therefore a quantitative measure 
of its degree of acidity or alkalinity. 

Solutions containing a weak acid (such as acetic acid HAc) and 
a salt of it which is a strong electrolyte (e.g. sodium acetate, 
NaAc) have interesting properties. The completely dissociated 
salt yields a large quantity of Ac - ions in the solution. From the 
equation of dissociation of the acid, 

[H + ][Ac-]/[HAc] = K, 

we find that the presence of excess Ac - ions in the solution causes 
a decrease in the number of H + ions, i.e. inhibits the dissociation 
of the acid. The concentration [HAc] of undissociated acid 
molecules is therefore practically equal to the total concentration 
of the acid (denoted by c a ). The concentration of Ac - ions, which 
are almost entirely supplied by the salt, is practically equal to the 
salt concentration (c s ). Thus [H + ] = Kcjc s , and the pH of the 
solution is 

pH = -log 10 [H+] = -\og 10 K + log 10 (c,/c a ). 

This depends only on the ratio of concentrations of the salt and 
the acid. Thus dilution of the solution, or the addition of small 
quantities of any other acids or alkalis, has practically no effect 
on the pH of the solution. A solution of this type whose pH 
remains constant is called a buffer solution. 

§91. Activation energy 

Hitherto we have considered only the state of chemical 
equilibrium, leaving aside the question of reaction mechanisms 
and rates. The calculation of the number of collisions of mole- 
cules in §88 served only to derive the conditions of equilibrium 
and, as already mentioned, may not correspond to the actual 
mechanism of the reaction. 



§91] 



ACTIVATION ENERGY 



267 



Let us now consider the rate at which a reaction occurs. 
Individual molecules can react with one another when they 
collide, but not all collisions bring about reactions: in reality, 
usually only a very small fraction of all collisions result in 
reactions between molecules. The explanation of this is as 
follows. 

In a reaction, the atoms of the colliding molecules are rearranged 
in a certain manner. For simplicity, let us assume that the 
reaction consists of a transfer of one atom from one molecule 
(A) to another molecule (B). The potential energy of this atom 
depends on its position with respect to the two molecules. The 
form of this energy as a function of a coordinate x along the "path 
of transition" of the atom is represented diagrammatically by a 
curve of the kind shown in Fig. 112. This diagram is, of course, 
highly schematic, since in reality the potential energy depends 
on several parameters (coordinates) and not on only one. What is 
important is not the precise variation of the potential energy but 
simply the fact that it has two minima corresponding to positions 
of the atom in each of the two molecules. These two positions are 
separated by a potential barrier. 




Fig. 112. 



A chemical reaction can occur only if the atom which is to be 
transferred between two colliding molecules has sufficient energy 
to traverse the barrier. In the majority of molecules, however, 
this atom has an energy equal to or close to the corresponding 
minimum. The molecule can react, therefore, with the transition 
A-» B, only if it has excess energy equal to U — U A (see Fig. 112). 
The ratio of the number of such molecules in the gas to the num- 



268 CHEMICAL REACTIONS 

ber which do not possess such energy is equal to the ratio of the 
Boltzmann factors (see §55): 

e -UolkT . £ -UAlkT — e ~(Uo-UA)lkT 

The energy U — U A is called the activation energy of the reaction 
concerned, and usually referred to one mole of the substance by 
multiplying U —U A by Avogadro's number: N (U —U A ) = E. 
Thus the number of molecules capable of reacting, and there- 
fore the reaction rate, are proportional to the activation factor 

e ~EIRT 

This is the principal factor in the temperature dependence of the 
reaction rate, and we see that the reaction rate increases very 
rapidly with temperature. 

If the reaction rate is denoted by v, it follows from the above 
that 

log e f = constant — E/RT, 

i.e. the logarithm of the reaction rate is a linear function of I IT. 
The slope of the straight line representing this function gives the 
activation energy E. 

The activation energy may have very different values for differ- 
ent molecular processes. For the majority of observable reactions 
it lies in the range from 10 to 1 50 kJ. 

When the temperature changes from a value T to a slightly 
different value T + AT, the change in the reaction rate is given by 
the formula 

log e U 2 - loge^i = logeivJVt) 



R(T + AT) RT RT 2 

For example, when E= 80 kJ, r = 300°K, AT = 10°, we find 
vjvi = 3. This is a typical increase in the reaction rate. For 
many kinds of reactions in gases and solutions, it is found that 
an increase of 10° in the temperature (in the range where the 



§91] ACTIVATION ENERGY 269 

reaction occurs at an appreciable rate) increases the reaction rate 
by a factor between two and four. 

The extent to which the reaction rate depends on the tempera- 
ture can be seen, for example, from the reaction 2HI — » H 2 + 1 2 , 
whose activation energy is 185 kJ. At 200°C the reaction still 
hardly occurs at all: the dissociation of an appreciable quantity 
of HI would take hundreds of years. At 500°C, the reaction is 
complete within seconds; yet even at this temperature only 
about one in 10 12 collisions between HI molecules results in 
dissociation. 

The necessity for a sufficiently high energy of the molecules is 
the principal reason for the low efficiency of collisions in produc- 
ing reactions. It is indispensable that the necessary excess energy 
should be concentrated on certain atoms or groups of atoms in 
the molecule; this fact also has a decisive influence in establishing 
the reaction rate. For reactions which involve complex molecules, 
there is also a geometrical factor: it is necessary that the reactive 
parts of colliding molecules should be in contact. 

Let us again consider Fig. 112. The difference U A — U B cor- 
responds to the difference of the internal energies of molecules 
A and B, i.e. the heat of reaction evolved in the exothermic 
reaction A — > B or absorbed in the endothermic reverse reaction 
B — » A . This difference is not directly related to the height of the 
potential barrier, i.e. there is no direct relation between the heat 
of reaction and the activation energy of the reaction. But there is 
a relation between the heat of reaction and the difference of the 
activation energies of the forward and reverse reactions. The 
diagram shows that the activation energies of the reactions 
A — > B and B — > A are U — U A and U —U B ; their difference is 
equal to the heat of reaction: 

(U - U B ) - (U - U A ) =U A - U B . 

As already mentioned in §88, reactions do not usually proceed 
exactly in accordance with the overall chemical equation: in 
reality, most chemical reactions have a more or less complex 
mechanism consisting of a number of simple elementary pro- 
cesses, the intermediate stages of the reaction, which are often 
difficult to determine. The reaction as it were selects the quickest 
path. The intermediate stages of the reaction must, of course, 



270 CHEMICAL REACTIONS [XI 

have the lowest possible activation energies; this is the funda- 
mental physical factor which determines the path of the reaction. 
The different stages may occur at very different rates. The rate 
of the overall process will evidently be determined mainly by 
the slowest of these intermediate stages, just as the speed of an 
assembly line can never be faster than the speed of the slowest 
operation. 

Decreasing the activation energy of the intermediate stages of 
a reaction is the basis of most processes of catalysis, i.e. the 
acceleration of reactions by adding to the reaction mixture another 
substance called a catalyst. This acceleration may be very great; 
reactions which otherwise practically do not occur at all often 
take place rapidly when a catalyst is present. The function of the 
catalyst is to participate in intermediate reactions in some way, 
while being restored to its original form as a result of the whole 
process. 

It should be emphasised that a catalyst can not displace the 
position of chemical equilibrium, which does not depend on how 
the reaction occurs. The only effect of the catalyst is on the rate at 
which equilibrium is established. 

§92. Molecularity of reactions 

All chemical reactions in gases or in weak solutions can be 
assigned to a number of types, depending on the number of 
molecules which must collide in order to bring about the reaction. 
Here, it must be emphasised, the true molecular processes which 
actually occur are meant. In the examples given below the 
reactions in fact occur in the way shown by the chemical equation. 
In most cases, however, this classification of reactions relates 
to the individual elementary stages of a complex reaction 
mechanism. 

Monomolecular reactions are those in which the molecules of 
the original substance decompose into two or more parts, for 
example the decomposition of ethyl bromide: 

C 2 H 5 Br^C 2 H 4 + HBr. 

There is no need for molecules to collide in order to bring about 
such reactions. Thus, as the substance decomposes, the reaction 
rate decreases linearly with its concentration. 



§92] MOLECULARITY OF REACTIONS 271 

In this sense reactions in weak solutions behave similarly 
when solvent molecules participate in addition to one molecule 
of solute, for example in the hydrolysis of cane sugar already 
mentioned above: 



cane sugar + H 2 — > glucose + fructose. 

This reaction in fact involves two molecules, but since there 
are plenty of water molecules round every sugar molecule through- 
out the reaction, the change in the reaction rate is due solely to 
the change in the concentration of dissolved sugar. 

Reactions in which two molecules yield two or more molecules 
are said to be bimolecular; for example, the reactions 

H 2 + I 2 ^2HI, 
N0 2 + CO^NO + C0 2 

are bimolecular in both directions. A collision between two 
molecules is necessary for such reactions to occur, and the rate 
is therefore proportional to the product of concentrations of the 
reacting substances (or to the square of the concentration, if the 
reaction involves two identical molecules). This type comprises 
the great majority of the elementary processes which make up the 
mechanism of complex reactions. 

Finally, trimolecular reactions are those in which three mole- 
cules take part to give two or more other molecules. These are 
comparatively few, because they can occur only if three mole- 
cules collide simultaneously, and such ternary collisions are, of 
course, much rarer than collisions between pairs of molecules. 

It is easy to determine the ratio of the numbers of ternary and 
binary collisions of molecules in a gas. We can say that ternary 
collisions of a given molecule are those which it undergoes while 
at the same time in the vicinity of a third molecule. Let V denote 
the total volume occupied by the gas, and b the total volume of the 
gas molecules. It is evident that the volume within which a mole- 
cule must be situated in order to be considered as in the vicinity 
of some other molecule is of the order of b, and the probability 
that the molecule is in the vicinity of another molecule is therefore 
b/V. The ratio of the numbers of ternary and binary collisions is 



272 CHEMICAL REACTIONS [XI 

consequently also of the order of b/V. This is a small quantity; 
for example, for air under standard conditions it is about 10~ 3 . 

The number of quaternary collisions is less than that of ternary 
collisions in the same ratio. Because of the extreme rarity of such 
collisions, chemical reactions of higher orders (quadrimolecular, 
etc.) do not occur in Nature. 

Some reactions which would appear to be bimolecular are in 
fact trimolecular. These are reactions in which two particles 
combine into one, for example 

H + H -> H 2 . 

If an H 2 molecule were formed by a collision of two H atoms, it 
would immediately dissociate again; the two colliding atoms can 
always move apart again. A stable H 2 molecule must have a 
negative internal energy. Thus two hydrogen atoms can form a 
stable molecule only when a further particle is present to receive 
the excess energy liberated in the formation of the molecule. 
This means that the reaction in question actually occurs only in a 
collision between three particles. 

It is interesting to note that even reactions which are clearly 
monomolecular sometimes behave as if they were bimolecular. 
In order to decompose, a molecule must have sufficient energy 
for its parts to overcome the potential barrier as they separate. 
An "activated" molecule of this kind has a definite "lifetime"; 
in a complex molecule, for example, the excess energy must be 
concentrated at the point where it is required for the decomposi- 
tion. Activated molecules are formed as a result of collisions 
between molecules in their thermal motion. In a sufficiently 
rarefied gas, where collisions are comparatively infrequent, 
activated molecules decompose more rapidly than fresh ones are 
formed. Under these conditions, the reaction rate is mainly deter- 
mined by the rate of the activation process, which requires colli- 
sions between molecules and is therefore a bimolecular process. 

§93. Chain reactions 

A characteristic feature of the mechanism of the majority of 
reactions is that fragments of molecules appear as intermediate 
products. These are individual atoms or groups of atoms, known 
as free radicals, which do not exist in a stable state. 



§93] CHAIN REACTIONS 273 

For example, in the decomposition of heated nitrous oxide gas 
(for which the formal equation is 2N 2 = 2N 2 + 2 ), the N 2 
molecules decompose thus: N 2 — > N 2 + 0, forming free atoms 
of oxygen, which then react with further molecules of nitrous 
oxide: O + N 2 -> N 2 + 2 . 

In this instance the intermediate particles (O atoms) disappear 
after the two component processes have occurred. There are 
many reactions, however, in which active intermediate products 
react continuously, and thus act as a kind of catalyst. 

This very important type of reaction mechanism may be 
illustrated by the formation of hydrogen bromide in a mixture of 
hydrogen and bromine vapour when the mixture is exposed to the 
action of light. This reaction in fact does not occur by collision 
of H 2 and Br 2 molecules, as would correspond to the chemical 
equation H 2 + Br 2 = 2HBr. Its true mechanism is as follows. 
Under the action of light, Br 2 molecules dissociate into two atoms: 

Br 2 -^ Br + Br. 

This is called chain initiation, and the bromine atoms formed act 
as active centres. These atoms, on colliding with H 2 molecules, 
react thus: 

Br + H 2 -^HBr + H. 

The resulting H atoms in turn react with Br 2 molecules: 

H + Br 2 ^ HBr + Br, 

again forming bromine atoms, which react with H 2 molecules, 
and so on. A continuous chain of successive reactions results, in 
which the Br atoms act as a kind of catalyst, being restored un- 
changed after the formation of two HBr molecules. This is called 
a chain reaction. The principles of the theory of chain reactions 
were worked out by N. N. Semenov and C.N. Hinshelwood. 

We see that, if active centres are formed in some way, the 
chain reaction will then proceed spontaneously, and could go to 
completion, one might think, without further external interaction. 
In reality, however, chain termination must also be taken into 
account. One active centre — a bromine atom in the above 



274 CHEMICAL REACTIONS [XI 

example — can cause the reaction of hundreds of thousands of 
hydrogen and bromine molecules, but it must eventually be lost, 
thus stopping the further progress of the chain reaction. This 
can occur, for example, by the recombination of two Br atoms 
to form a Br 2 molecule. It has been mentioned in §92, however, 
that such a combination of two atoms to form a stable molecule 
can occur only by a ternary collision. This mechanism of chain 
termination therefore becomes important only at high pressures, 
when ternary collisions in the gas are fairly frequent. 

Another mechanism of chain termination consists in the loss 
of active centres when they strike the walls of the reaction vessel. 
This is of importance at low pressures, when the active centres 
can move quite easily through the gas. 

On the other hand, there exist reactions in which chain 
branching occurs. For example, the combustion of hydrogen in 
a detonating mixture of hydrogen and oxygen occurs (at high 
temperatures) essentially as follows. By an external interaction 
(e.g. passage of a spark) a chain is initiated: 

H 2 + 2 ^ 20H. 

The resulting active centres (OH radicals) react with H 2 molecules 
to give water: 

OH + H 2 ^H 2 + H. 

The H atoms thus formed then react as follows: 

H + 2 -+OH + 0, 
+ H 2 ^OH + H. 

These reactions not only yield water but also increase the 
number of active centres H, O and OH (unlike the reaction of 
HBr formation, where the number of free H and Br atoms did 
not increase). 

If the increase in the number of active centres by chain branch- 
ing outweighs the termination of chains, then this number grows 
very rapidly (in geometric progression), and the reaction is 
thereby accelerated into an explosion. 



§93] CHAIN REACTIONS 275 

This chain mechanism of explosion has the feature that it can 
in principle develop even at constant temperature. Another 
important explosion mechanism is the thermal mechanism, 
which results from the marked temperature dependence of the 
reaction rate. When heat is rapidly evolved in an exothermic 
reaction, the rate of removal of heat may be insufficient, and 
consequently the reaction mixture will be heated, thus leading to 
a progressive spontaneous acceleration of the reaction. 



CHAPTER XII 

SURFACE PHENOMENA 



§94. Surface tension 

So far we have discussed thermal properties and phenomena 
which occur throughout volumes and affect the whole mass of a 
body. The existence of free surfaces of bodies brings about the 
existence of a separate class of surface phenomena or capillarity 
effects. 

Strictly speaking, any body is in an external medium, such as the 
atmosphere, and not in a vacuum. Thus we should speak not merely 
of the surfaces of bodies but of interfaces between two media. 

In surface phenomena, only those molecules which are actually 
at the surfaces of bodies are involved. If the bodies are not very 
small, the number of such molecules is very small in comparison 
with the total number of molecules in the bodies. For this reason, 
surface phenomena are usually of minor importance, but they 
become significant in small bodies. 

The molecules which are in a thin layer adjoining the surface 
are in conditions different from those within the body. The latter 
are surrounded by similar molecules on all sides, whereas the 
molecules near the surface have similar molecules on one side 
only. This has the result that the energy of the molecules in the 
surface layer is different from that of the molecules within the 
body. The difference between the energy of all the molecules 
(in both media) which are near the surface and the energy which 
they would have within the bodies is called the surface energy. 

It is evident that the surface energy is proportional to the area 
S of the interface: 

t^surf = OiS. 

The coefficient a depends on the nature and state of the media in 
contact; it is called the surface tension. 

276 



§94] 



SURFACE TENSION 



277 



As we know from mechanics, forces always act so as to bring 
a body to a state of minimum energy. In particular, the surface 
energy tends to take its least possible value. Hence it follows that 
the coefficient a is always positive, since otherwise the media in 
contact could not exist separately; their interface area would 
tend to increase without limit, i.e. the two media would tend to 
mix. 

Conversely, since the surface tension is positive, the interface 
between two media must always tend to contract. This is the 
reason why droplets of liquid (and gas bubbles) tend to be 
spherical: for a given volume, the sphere is the figure with the 
least area. This tendency is opposed by the force of gravity, but 
for small droplets its effect is slight and their shape is almost 
spherical. In conditions of weightlessness this will be the shape 
of any free mass of liquid. Such conditions may be simulated in 
a well-known experiment with a spherical drop of oil floating 
within a mixture of alcohol and water having the same specific 
gravity. 




Fig. 113. 



The surface-tension force is shown by the following simple 
example. Let us imagine a film of liquid supported on a wire 
frame of which one side (of length /) is movable (Fig. 113). 
Because the surface tends to contract, the wire is subject to a 
force which can be directly measured on the movable part of the 
frame. By the general laws of mechanics, this force is the deriv- 
ative of the energy (in this case, of the surface energy) with 
respect to the coordinate x in the direction in which the force 
acts: 



F = — dU SUTf ldx = — a dS/dx. 



278 SURFACE PHENOMENA [XII 

But the area of the film is S = Ix, and therefore 

F = -al. 

This is the force on a segment / of the frame due to the surface- 
tension force on one side of the film; since the film has two sides, 
the force on the segment / is twice this value. The minus sign 
shows that this force is directed into the surface of the film. 

Thus the line bounding the surface of the body (or any part of 
this surface) is subject to forces perpendicular to the line and 
tangential to the surface, directed into the surface. The force 
per unit length is equal to the surface tension a. 

The dimensions of a follow from its definition, and may be put 
in various forms: energy per unit area, or force per unit length, 

[a] = erg/cm 2 = dyn/cm. 

It is clear from the above that, in stating the value of the surface 
tension, it is necessary to state which two media are in contact. 
The term "surface tension" is often applied to a liquid (without 
specifying any other medium) to denote the surface tension 
between the liquid and its vapour. This quantity always decreases 
with increasing temperature and becomes zero at the critical 
point, where the difference between liquid and vapour ceases to 
exist. 

The following list gives the surface tension (in erg/cm 2 ) between 
various liquids and air: 



Water (20°C) 73 


Mercury (20°C) 


480 


Ethyl ether (20°C) 17 


Gold(1130°C) 


1100 


Benzene (20°C) 29 







Liquid helium has a very low surface tension at an interface with 
its vapour, only 0-35 erg/cm 2 (near absolute zero). 

A surface tension also exists, of course, at the surfaces of 
solids, but here its effect is very slight under ordinary conditions: 
the comparatively weak surface forces cannot change the shape 
of a solid body. A direct measurement of the surface tension of 
solids is therefore very difficult, and there are no reliable data 
as to its values. 



§95] ADSORPTION 279 

The surface tension of an anisotropic body (a crystal) must be 
different on different faces, since in general the atoms are differ- 
ently arranged on different faces. For this reason, if a crystal 
could freely change shape under the action of external forces, 
it would not become spherical as an isotropic body (a liquid) 
would, where the surface tension is everywhere the same. It 
can be shown that the equilibrium shape of a crystal under these 
conditions is a very curious one, consisting of a relatively small 
number of plane faces, which, however, do not meet at angles 
but are joined by rounded regions. 

This phenomenon may be observed, for example, on prolonged 
heating (at about 750°C) of spheres of rock salt cut from single 
crystals. The high temperature assists the atoms in "creeping" 
from one point on the surface to another, and in consequence the 
sphere is converted into a figure of the kind described. 

§95. Adsorption 

Many surface phenomena come under the heading of adsorp- 
tion, which consists in the adhesion of substances on the surfaces 
of solids and liquids (the latter being then called adsorbents). 
Adsorption can take place from gases or liquids, and a solute 
may be adsorbed from solution. For example, many gases are 
adsorbed on the surface of carbon, silica gel, or the majority of 
metals; carbon adsorbs various organic compounds from solution. 
The degree of adsorption is described by the surface concentra- 
tion, which is the quantity of the substance per unit area of the 
surface of the adsorbent. 

Adsorption phenomena are widely found in Nature, and play 
an important part in technology. In order to adsorb a large 
quantity of a substance, we must evidently use substances which 
have the maximum area for a given mass, such as porous or 
finely powdered materials. To describe this property of adsor- 
bents, we use the specific area, which is the area per unit mass of 
the substance. In good adsorbents, such as specially prepared 
porous carbons, it reaches hundreds of square metres per gram. 
Such large values of the specific area are not surprising if we 
consider how rapidly the surface area increases when a body is 
permeated by pores or is finely crushed. For example, 1 cm 3 of 
material in spheres of radius r cm will have a total area of 3/r cm 2 , 
and when r ~ 10~ 6 this amounts to hundreds of square metres. 



280 SURFACE PHENOMENA [XII 

The concentration of adsorbed gas depends (at a given tem- 
perature) on the gas pressure over the surface of the adsorbent. 
This dependence is shown by a curve, called an adsorption iso- 
therm, of the form shown in Fig. 114. The surface concentration 
at first increases rapidly with pressure. As the pressure continues 
to rise, the concentration increases more slowly, and finally 
reaches a limit or saturation value. Experiment shows that the 
saturation of adsorption corresponds to a more or less dense 
occupation of the adsorbent surface by a single layer of adsorbed 
molecules (called a monomolecular layer). 




A very important property of adsorption is the change which it 
causes in the surface tension at the interface between media. 
Usually the surface of a liquid is concerned. Adsorption always 
reduces the surface tension, since otherwise adsorption would 
not occur. Here again there is a tendency to reduce the surface 
energy: this reduction can be achieved not only by decreasing 
the surface area but also by changing the physical properties of 
the surface. Because of their effect on the surface tension, sub- 
stances which can be absorbed (on the surface of a given liquid) 
are said to be surface-active. On water, for example, various 
soaps are surface-active. 

The total quantity of a substance which can be absorbed on 
the surface of a liquid is very small. Thus even small quantities 
of surface-active substances accumulating on the surface of a 
liquid may considerably affect its surface tension. The surface 
tension of a liquid is very sensitive to impurities: for example, 
even very small quantities of soap can reduce the surface tension 
of water by a factor of more than three. 

Adsorbed monomolecular films on the surfaces of liquids are 
a very curious physical phenomenon, forming as it were a two- 



§95] ADSORPTION 281 

dimensional state of matter, in which the molecules are distributed 
over a surface in two dimensions and not over a volume in three 
dimensions. In this state there can exist various phases, "gas", 
"liquid" and "solid", exactly analogous to three-dimensional 
phases. 

In a "gaseous" film the adsorbed molecules have a compara- 
tively rarefied distribution on the surface of the liquid and can 
move freely on it. In "liquid" and "solid" films the molecules are 
close together, either retaining some freedom of relative motion 
(so that a liquid film can "flow"), or so firmly held together that 
the film behaves as a solid. Liquid and solid films may be aniso- 
tropic, forming two-dimensional analogues of liquid and solid 
crystals; in the liquid film we have a regular orientation of mole- 
cules on the surface of the adsorbent, and in the solid film a type 
of two-dimensional crystal lattice with a regular configuration of 
molecules. It is noteworthy that such anisotropic films may occur 
at an interface between two isotropic media, a liquid and a gas. 

These effects are very well illustrated by the monomolecular 
films formed on a water surface by insoluble complex organic 
acids, alcohols etc., whose molecules form long hydrocarbon 
chains with — COOH, —OH, etc., groups at one end. These 
groups are strongly attracted by the water molecules and, as it 
were, dissolve in the surface layer of the water, but cannot carry 
the whole molecule into the liquid; part of the molecule remains 
above the surface. Thus a liquid or solid film forms a kind of 
forest of closely packed molecules with their ends immersed in 
water. 

The surface tension a when the water surface is covered with 
a film is less than its value a for the clean surface. The difference 
a — a can be measured directly from the force acting on a barrier 
floating freely on the surface of the water and separating the film 
from the clean surface. The film exerts a force a (into the film) 
per unit length of this barrier, and there is an opposite force a 
exerted by the clean surface. Since a > a, the result is that the 
film repels the barrier with a force 

Aa = a — a 

per unit length. This force may be regarded as the pressure of 
the film. At a given temperature, it is a definite function of the 
area S of the film (formed by a given quantity of the adsorbed 



282 



SURFACE PHENOMENA 



[XII 



substance), just as the pressure of an ordinary body is a function 
of its volume. 

For a gaseous, rarefied film (with n molecules in the area S), 
this relation is given by 

Aa = nkT/S, 



which is similar to the equation of state of an ideal gas 
(p = NkTjV). When the film is compressed (i.e. when its area 
S decreases), there occurs at a certain value of Aa a phase transi- 
tion to a continuous liquid or solid film. On the curve of Aa as 
a function of S this transition corresponds to a horizontal section, 
which is entirely similar to that for the ordinary transition between 
vapour and liquid on the isotherms which show the relation 
between the pressure p and the volume V (§70). 

§96. Angle of contact 

At the edge of a liquid surface in a vessel we have three media 
in contact: the solid wall (medium 1 in Fig. 115), the liquid (2) 
and the gas (3). Let us consider the capillary effects which occur 
at such a boundary. 





Fig. 115. 

Three forces of surface tension act on the line along which all 
three media are in contact; this line intersects the plane of the 
diagram at O. Each force is directed tangentially inwards along 
the interface between the two media, as shown by the arrows in 
the diagram. The magnitudes of the forces per unit length of 
the line of contact are equal to the respective surface tensions 
« 12 , «i 3 , a 23 . The angle between the surface of the liquid and the 
solid wall will be denoted by and is called the angle of contact. 



§96] ANGLE OF CONTACT 283 

The surface of the liquid takes a form such that the resultant 
of the three forces a 12 , ot 13 , a 23 has no component along the wall 
of the vessel (the component perpendicular to the wall is balanced 
by the reaction of the wall). Thus the condition of equilibrium of 
the liquid at the wall is 

«13 = «12 + «23COS 6, 

whence 

cos = (a 13 — at l2 )la 23 . 

We see that the angle of contact depends only on the nature of 
the three media in contact (through the surface tensions at their 
interfaces); it does not depend on the shape of the vessel or on the 
force of gravity acting on the bodies. It must be remembered, 
however, that the surface tensions, and therefore the angle of 
contact, are very sensitive to the state of cleanliness of the 
interfaces. 

If a 13 > a l2 , i.e. if the surface tension at the interface between 
the solid wall and the gas is greater than that at the interface 
between the wall and the liquid, then cos > and the angle 
is acute. In other words, the edge of the liquid is raised, and its 
surface or meniscus is concave (Fig. 115a). The liquid is then 
said to wet the solid surface. If a drop of such a liquid is placed 
on the surface of the solid, it "flows" to some extent over the 
surface (Fig. 116a). 

If, on the other hand, a 13 < a 12 , then cos < and is obtuse; 
the edge of the liquid is depressed and its meniscus is convex 
(Fig. 1 15b). In this case we say that the liquid does not wet the 
solid. For example, the angle of contact of mercury on glass is 
about 150°, and that of water on paraffin wax is about 105°. 
Drops of such liquids, when placed on the solid surface, appear 
to contract so as to reduce the area of contact with the surface 
(Fig. 116b). 



(a) 

Fig. 116. 




284 SURFACE PHENOMENA [XII 

Since the cosine of an angle cannot exceed unity in absolute 
magnitude, it is seen from the formula derived above for cos 6 
that in any actual case of stable equilibrium between the liquid 
and the wall the condition 

|«13 — «12| ^«23 

must hold. If «i2, a 13 , a 23 are taken to be the surface tensions for 
each pair of media alone, in the absence of the third medium, this 
inequality may certainly prove to be violated. In reality, we must 
remember, the third substance may be adsorbed on the interface 
between the other two and thus lower the surface tension, so 
that the resulting values of a are such as to satisfy the foregoing 
condition. 

The concepts of wetting and non- wetting in the sense explained 
above must be distinguished from that of complete wetting, 
which refers to the condensation of a vapour on the surface of a 
solid. As we know, the condensation of a vapour to a liquid 
is brought about by the action of the van der Waals forces of 
attraction between molecules. These forces, however, can be 
exerted on a molecule in the vapour not only by similar mole- 
cules but also by the molecules of a solid. Let us suppose that the 
attraction forces from the solid are stronger than those in the 
liquid itself. In such a case, the presence of the solid surface will 
clearly bring about a partial condensation of the vapour even in 
conditions where the vapour is unsaturated and would therefore 
otherwise be stable. A thin film of liquid forms on the surface of 
the solid. The thickness of such a film cannot be great, of course; 
its order of magnitude is determined by the range of action of 
the van der Waals forces and may be from 10~ 7 to 10~ 5 cm. As 
the vapour approaches saturation the film becomes thicker. This 
effect is called complete wetting of the solid by the liquid. For 
example, carbon tetrachloride (CC1 4 ) completely wets many 
surfaces, including that of glass. 

[The difference between this phenomenon and adsorption 
should be emphasised: here we are discussing a very thin but 
still "macroscopic" layer of liquid, whereas an adsorbed film 
consists of individual molecules distributed over the surface.] 

The edge of a liquid which completely wets the walls of a vessel 
passes continuously into the film on the wall. Thus in this case 



§97] CAPILLARY FORCES 285 

there is no finite angle of contact. We may say that complete 
wetting corresponds to zero angle of contact. A drop of such a 
liquid placed on the surface will spread completely over it. 

More complex types of wetting are in principle possible, 
depending on the nature of the van der Waals forces exerted 
by the solid. For instance, a case is possible where the vapour 
condenses into a liquid on the solid surface but the thickness of 
the resulting film can not exceed a certain limiting value. If the 
surface is already covered with such a film, a further drop of 
liquid placed on it will not spread completely, but will remain 
isolated, though highly flattened, with a very small but finite 
angle of contact. This seems to occur for water on clean glass; 
the maximum film thickness is about 10 -6 cm, and the angle of 
contact is probably less than one degree. 

§97. Capillary forces 

It has several times been mentioned that, in a state of equi- 
librium, the pressures of bodies in contact must be equal. In 
reality this statement is true only in so far as capillary effects 
are neglected. When the surface tension is taken into account, 
the pressures in adjoining media are in general different. 

Let us consider, for example, a drop of liquid in air. The 
tendency of the drop surface to decrease causes a contraction of 
the drop and therefore an increase in its internal pressure. The 
pressure of the liquid in the drop therefore exceeds the pressure 
of the surrounding air. The difference between them is called the 
surface pressure and will be denoted by p surf . 

To calculate this quantity, we note that the work done by the 
surface forces in reducing the surface area of the drop by dS is 
equal to the corresponding decrease a dS in the surface energy. 
This work can also be written as p sur fdV, where dV is the change 
in the volume of the drop; thus 

adS = p SUT fdV. 

For a spherical drop of radius r, S = 4nr 2 and V= 4-7^/3; 
substitution in the above equation then gives the following 
expression for the surface pressure: 

Psurf = 2a/r. 



286 



SURFACE PHENOMENA 



[XII 



This formula applies also, of course, to a bubble of gas in a 
liquid. The higher pressure always occurs in the medium towards 
which the interface is concave. When r -» °°, the surface pressure 
tends to zero. This is in accordance with the fact that for a plane 
interface the pressures in the adjoining media must be the same; 
it is evident that the tendency of the surface to contract will not 
lead to any force into either medium in this case. 

We may also derive a formula for the surface pressure in a 
cylindrical mass of liquid. In this case S = lirrh, V= nr 2 h (where 
r is the radius and h the height of the cylinder), and substitution 
in the equation p surt dV = adS gives 

Psurf = air. 

These simple formulae enable us to solve a number of problems 
relating to capillarity effects. 



-v- 
0t 



^ 




(a) 



(b) 



Fig. 117. 



Let us imagine two parallel flat plates (shown in cross-section 
in Fig. 1 17) between which is a thin layer of liquid. The lateral 
surface of the liquid is in contact with air. If the angle of contact 
is acute, the meniscus of the liquid is concave and the pressure 
within the liquid is less than the air pressure; the atmospheric 
pressure acting on the plates will therefore tend to bring them 
together and they appear to attract each other; if the angle of 
contact is obtuse, the meniscus is convex and the layer of liquid 
appears to push the plates apart. When the space between the 
plates is sufficiently narrow, any small section of the meniscus 
may be regarded as part of a cylindrical surface of some radius r. 



§97] CAPILLARY FORCES 287 

A simple construction (Fig. 1 17b) shows that x = 2r cos 6, where 
x is the distance between the plates. The pressure in the liquid is 
less by p smt = a/r — (2a/x) cos 0, where a is the surface tension 
between the liquid and the air. The force of attraction F between 
the plates is found by multiplying this quantity by the area of 
contact S between the liquid and each plate: 

F = (2aS/x) cos 0. 

We see that this force is inversely proportional to the distance 
between the plates. When the distance is small, the force may be 
very large; for example, plates separated by a film of water one 
micron thick are attracted together by a pressure of about 
1 -5 atm. 

Let us next consider the well-known capillary rise (or fall) of 
a liquid in a narrow tube immersed in the liquid. When the 
meniscus is concave (acute angle of contact) the pressure of the 
liquid in the tube is less than that of the adjacent air by an 
amount p surf . The atmospheric pressure on the surface of the 
liquid in the vessel therefore causes the level of the liquid in the 
tube to rise until the weight of the column of liquid balances 
the extra pressure: p smt = pgh, where p is the density of the liquid. 
The surface of the meniscus in a narrow tube may be regarded as 
part of a sphere whose radius r is related to the radius a of the 
tube by a = r cos 0. Then p surf = 2ajr = (2a/ a) cos and the 
height to which the liquid rises is 

h — lajgpr 
= {lalgpa) cos 0. 

For a convex meniscus, this formula gives the depth to which 
the liquid sinks. 

The surface tension and the density of the liquid appear in the 
above formula in the combination a/pg. The quantity V(2a/pg) 
has the dimensions of length and is called the capillary constant. 
It plays an important part in all phenomena which occur under 
the combined action of surface-tension forces and gravity. The 
capillary constant of water at 20°C is 0-39 cm. 

Various effects of capillary forces are the basis of methods 
for measuring surface tension. For example, the size of drops of 



288 SURFACE PHENOMENA [XII 

liquid flowing slowly from a narrow tube is determined by the 
equilibrium between the weight of the drop and the surface 
tension around its "neck"; thus a measurement of the weight 
of the drop (by counting the number of drops formed by a given 
quantity of liquid) enables us to determine a. Another method 
is based on measurement of the surface pressure within a gas 
bubble of given radius; this is effected by measuring the additional 
pressure necessary to expel a bubble of air from the end of a tube 
immersed in the liquid. 

§98. Vapour pressure over a curved surface 

The influence of capillary forces causes some changes also in 
the properties of equilibrium between a liquid and its saturated 
vapour. It has been stated above that the saturated vapour 
pressure is a definite function of temperature. In reality, this 
pressure depends also on the shape of the liquid surface above 
which the vapour is situated. The dependence is admittedly very 
slight, and can be of importance only for small bodies (e.g. 
droplets of liquid). 

The nature and amount of this dependence are easily deter- 
mined by again considering the capillary rise (or fall) of a liquid 
and supposing that the space above the liquid in the vessel and 
in the tube is filled with saturated vapour. Since the gas pressure 
decreases with increasing height, it will evidently be smaller 
above a liquid which has risen (and larger above a liquid which 
has fallen) than above the flat surface of the liquid in the vessel. 
Comparing this with the shape of the meniscus in the tube in 
the two cases, we conclude that the saturated vapour pressure 
above a concave liquid surface is less (and above a convex 
surface greater) than above a flat surface. The similarity of this 
argument to the derivation of Raoult's law in §81 should be 
noted. 

If h is the height of the capillary rise, the decrease in the 
saturated vapour pressure is Ap = p vap gh. We have seen in §97 
that h = lalpirg, where p t is the density of the liquid and r the 
radius of the sphere of which the meniscus forms part. Thus 
we have 

An _ Z<% Pvap 

r Pi' 



§99] THE NATURE OF SUPERHEATING AND SUPERCOOLING 289 

The decrease in the saturated vapour pressure above a concave 
surface causes what is called capillary condensation, the deposi- 
tion of a liquid in a porous body from a vapour which under 
ordinary conditions would not be saturated. If the liquid wets 
the body concerned, concave menisci of liquid are formed in the 
pores (which act as very fine capillaries), and the vapour may then 
be supersaturated even at a comparatively low pressure. 

When the liquid surface is convex, the same formula for Ap 
gives the amount by which the vapour pressure exceeds its value 
over a flat surface. We see that the saturated vapour pressure 
above a drop of liquid increases with decreasing radius of the 
drop. 

Let us imagine a vapour containing a large number of liquid 
droplets of various sizes. It may happen that the vapour is super- 
saturated with respect to the larger droplets but unsaturated with 
respect to the smaller ones. Then the liquid which evaporates 
from the smaller drops will condense on the larger ones, which 
as it were "consume" the small drops. 

§99. The nature of superheating and supercooling 

The most important consequence of the dependence of the 
saturated vapour pressure on the size of a drop is that it gives 
an explanation of supersaturation of a vapour— the continued 
existence of the gaseous state under conditions such that the 
substance should become liquid. 

A supersaturated vapour over the surface of the liquid will, 
of course, condense immediately, but if the vapour is not in 
contact with the liquid the condensation is impeded by the fact 
that it must begin with formation of small droplets in the vapour. 
A vapour supersaturated with respect to a flat liquid surface 
may still be unsaturated with respect to such droplets, which 
are then unstable and evaporate again as soon as they are 
formed. Only if a liquid drop happens to be formed in the vapour 
which is so large that the vapour is also supersaturated with respect 
to the drop will such a drop continue to exist and the vapour 
continue to condense on it; the drop will act as a nucleus 
of the new phase. Spontaneous formation of such nuclei in 
completely pure vapour can occur only by random thermal 
fluctuations, and this is in general a very unlikely occurrence. 
Its probability decreases for increasing values of the "critical" 



290 SURFACE PHENOMENA [XH 

radius of the drop, i.e. the minimum radius which gives stability. 
As the degree of supersaturation increases, the "critical" radius 
becomes smaller and the formation of nuclei becomes easier. 
When this quantity reaches values of the order of molecular 
dimensions, the creation of special nuclei is essentially unneces- 
sary and further supersaturation of the vapour is impossible. 

The condensation of a supersaturated vapour is assisted by 
the presence of a solid surface in contact with it which is wetted 
by the liquid in question. Small droplets which are deposited 
on such a surface spread somewhat and their surfaces become 
less curved. Thus such drops can easily become centres of further 
condensation. Condensation occurs with particular ease on a 
surface which is completely wetted by the liquid, since the 
drops disperse over the whole of such a surface. 

Under ordinary conditions the vapour is not completely pure, 
and various small dust particles present in it act as centres of 
condensation, by forming solid surfaces which are wetted by the 
liquid. Thus, in order to achieve a considerable degree of super- 
saturation, a careful removal of all contamination from the 
vapour is necessary. 

Charged particles (ions) strongly attract the vapour molecules, 
and consequently small droplets immediately form around them 
and act as centres for further condensation; thus charged particles 
create particularly favourable conditions for the condensation of 
the vapour. This phenomenon is, in particular, the basis of the 
cloud chamber used for the observation of the paths of fast 
ionising atomic or nuclear particles. 

We have given a detailed discussion of the reasons for the 
occurrence of the metastable state of supersaturated (super- 
cooled) vapour. These reasons are in fact general and account 
also for the "delay" in other phase transitions. The formation 
of a new phase within a previously existing one must begin with 
the formation of small inclusions or nuclei of the new phase. 
For example, the conversion of a liquid into a vapour must begin 
with the appearance in the liquid of small bubbles of vapour; 
the solidification of a liquid, with the appearance in it of crystal 
nuclei, and so on. 

The additional surface energy at the boundary of such a nucleus 
makes its formation energetically unfavourable unless it is suffi- 
ciently large. Here we have a competition between two opposing 



§100] COLLOIDAL SOLUTIONS 291 

factors. The formation of a new interface between two phases 
involves the absorption of the surface energy, but when the sub- 
stance enter's a new phase there is a gain in volume energy. The 
latter quantity increases with increasing size of the nucleus 
more rapidly than the former quantity, and ultimately outweighs 
this. We may say that the formation of a nucleus of a new phase 
requires the traversing of a "potential barrier" due to the surface 
energy, and this is possible only for a sufficiently large nucleus. 

There is one phase transition which in this respect appears to 
form an exception to the general rule, namely the melting of 
crystals. When crystals are heated in an ordinary manner, super- 
heating is never observed. This, however, is simply because the 
surface of any crystal is completely wetted by the liquid formed 
when it melts. Thus the liquid droplets formed on the surface 
of the crystal spread over it, and surface tension does not act to 
prevent melting. 

Superheating of crystals can occur if the crystal is artificially 
heated from the inside instead of from the outside. For instance, 
when a current is passed through a single-crystal rod of tin 
with intensive air cooling of the exterior, the temperature within 
the rod is higher than that on its surface, and the interior of the 
crystal can then be superheated by one or two degrees before 
ordinary melting begins at its surface. 

§ 1 00. Colloidal solutions 

Sometimes a substance which does not dissolve in a given 
liquid can be distributed in it in the form of very fine particles, 
although these still contain a very large number of molecules. 
In this case the finely divided or dispersed substance is called 
a disperse phase, and the medium in which it is distributed is 
called the dispersion medium. If the size of the particles is of 
the order of 10~ 4 to 10 -2 cm, such a mixture is called a suspension 
or emulsion according as the particles are of a solid or a liquid; 
for example, milk is an emulsion of fat in water. 

When the particles are even smaller (10 -7 to 10 _5 cm, or 10 
to 10 3 A), the mixture is called a colloidal solution or sol. The 
characteristic property of these solutions is the size of the 
particles of the disperse phase rather than the number of mole- 
cules in each particle. For example, in a colloidal solution of gold 
in water, each particle is of size 100-500 A and contains millions 



292 SURFACE PHENOMENA [XII 

of gold atoms, but in solutions of such complex substances as 
proteins each colloid particle may contain only one molecule. 

The dispersion medium may be either a liquid or a gas. For 
example, colloidal solutions in air (aerosols) may be smokes, 
mists or fogs. The most important colloidal solutions, however, 
are those in liquids, in particular those in water (hydrosols). 
For example, the majority of substances concerned in the 
constitution of plants and animals are present in them as liquid 
colloidal solutions. 

Many kinds of substance are able to form sols: many organic 
compounds with large molecules (proteins, starch, gelatine, 
etc.), silicic acids, aluminium hydroxide, etc. Sols of some metals 
can also be obtained, for example of gold in water. 

Because of the high degree of dispersion of the disperse phase, 
the total surface area of its particles is extremely large, and 
surface phenomena therefore have a very important effect on the 
properties of colloidal solutions. 

Since the surface tension tends to reduce the area of the inter- 
face, the particles of the disperse phase have a tendency to 
combine and be precipitated from the solution as a dense mass. 
This tendency is counteracted by the forces of electrical repul- 
sion: the particles of the disperse phase in a colloidal solution 
are always electrically charged, and all the charges are of the 
same sign (which may be either positive or negative). Only this 
fact prevents the particles from coalescing and being precipitated. 

Colloidal particles are charged either because of the electrolytic 
dissociation of their molecules or by adsorption of ions from the 
surrounding fluid. When an electrolyte is added to a colloidal 
solution, the ions of the electrolyte may cancel the charge on the 
colloidal particles and render them electrically neutral. This 
brings about the precipitation or coagulation of the colloidal 
solution. Coagulation of colloids may also be effected by other 
means, for instance by heating. 

Colloidal solutions may be divided into two groups as regards 
stability. Some colloidal solutions are a stable state of matter 
and can be precipitated only with difficulty. These are called 
lyophilic colloids, and include hydrosols of proteins, gelatine, 
silicic acids, and other substances. When a lyophilic colloidal 
solution is coagulated it often becomes a jelly-like mass called 
a gel. This contains not only the substance from the disperse 



§100] COLLOIDAL SOLUTIONS 293 

phase but also a considerable quantity of the solvent (water, 
etc.). A gel is a kind of irregular network of solute particles, 
enclosing solvent molecules. A typical feature of the conversion 
of a lyophilic sol into a gel is that it is reversible: under appro- 
priate conditions a gel may absorb a sufficient quantity of solvent 
to become a sol again. 

Colloidal solutions of the other group form a metastable state 
of matter and are very easily precipitated. These lyophobic 
colloids include, for example, colloidal solutions of metals in 
water. The coagulation of lyophobic colloids is accompanied 
by the formation of a dense precipitate, and is an irreversible 
process; the precipitate can not be so easily converted into a 
solution again. 



CHAPTER XIII 

MECHANICAL PROPERTIES OF 
SOLIDS 



§101. Extension 

The work done on a liquid or gas depends only on the change 
in its volume, and not on the change in shape of the vessel 
containing it. Liquids resist change in volume but not change 
in shape. This property is the reason for Pascal's law in liquids, 
which states that the pressure transmitted by a liquid is the same 
in all directions: if, for instance, a liquid is compressed by a 
piston, the same pressure will be exerted by the liquid on every 
wall of the vessel. The pressure force acting on the liquid and 
transmitted by the liquid is always at right angles to the walls: 
a force tangential to the surface, and not capable of being com- 
pensated because a liquid offers no resistance to a change of shape, 
can not exist in equilibrium conditions. 

Solids, on the other hand, resist both change in volume and 
change in shape; they resist, therefore, any deformation. Work 
must be done even in order to change the shape alone of a solid, 
without altering its volume. We may say that the internal energy 
of a solid depends on its shape as well as on its volume. In con- 
sequence, Pascal's law does not apply to solids: the pressure 
transmitted by a solid is different in different directions. The 
pressures which occur in a solid when it is deformed are called 
elastic stresses. Unlike the pressure in a liquid, the elastic-stress 
force in a solid may be in any direction relative to the area on 
which it acts. 

The simplest type of deformation of a solid is extension. This 
occurs in a thin rod (Fig. 1 18a) of which one end is fixed, when a 
force F tending to stretch the rod is applied to the other end. (If 
the force F is in the opposite direction we have a compression.) 
It may be noted that fixing in a wall is, by the law of action and 

294 



§101] 



EXTENSION 



295 



reaction, equivalent to applying to the fixed end a force equal and 
opposite to the force acting on the free end (Fig. 1 1 8b). 

The elastic stresses in the rod are determined by the value F/S 
of the extending force per unit area of the cross-section S of 
the rod; let this value be p. The stresses are clearly constant 
along the length of the rod, and thus the same stretching stress 
p is exerted on each element of length of the rod by the adjoining 
parts of the rod (Fig. 1 1 8b). It is therefore clear that each unit 



■///////////A 



P 
(a) 



R 



(b) 



Fig. 118. 



length of the rod undergoes the same extension, and the total 
increase 8/ in the length of the rod is proportional to this length. 
Thus the relative elongation 

\ = 8/// 



(where / is the length of the rod before deformation) is indepen- 
dent of the length of the rod, and is clearly a measure of the degree 
of deformation undergone by each part of the rod. 

Because of the high strength of solids, the deformations which 
they undergo when subjected to external forces are usually small. 
That is, the relative changes in size of solid bodies are small, and 
in the case of the extension described above the relative elonga- 
tion is small. Such deformations may be assumed to be propor- 
tional to the stresses which cause them, and therefore to the 
magnitude of the external forces applied. This is called Hooke's 
law. 

For extension, Hooke's law implies that the relative elongation 



296 MECHANICAL PROPERTIES OF SOLIDS [XIII 

A is proportional to the tensile stress p. This relation is usually 
written 

\ = plE, 

where the coefficient E is a property of the material and is called 
Young's modulus. The relative elongation A is evidently a 
dimensionless quantity, and the modulus E therefore has the 
dimensions of/?, i.e. those of pressure. 

As examples, the values of Young's modulus (in millions of 
bars) for a number of materials are as follows: 

Iridium 5-2 Quartz 0-73 

Steel 2-0-2-1 Lead 0-16 

Copper 1-3 Ice (-2°C) 0-03 

Young's modulus, however, does not completely describe the 
properties of a body with respect to deformation (its elastic pro- 
perties). This is clear even for an extension. The reason is that 
longitudinal stretching of a rod involves a decrease in its trans- 
verse dimensions: the rod becomes thinner at the same time as 
its length increases. The value of Young's modulus enables us to 
calculate the relative elongation of the rod (for a given stress), 
but does not suffice to determine the transverse contraction. 

The relative decrease in the transverse dimensions of the rod 
is also proportional to the tensile stress p, and therefore to the 
relative extension A. The ratio of the relative transverse contrac- 
tion of the rod to its relative elongation is a quantity characteristic 
of any given material and is called Poisson's ratio, denoted by or. 
Thus the relative transverse contraction (e.g. the relative 
decrease in the diameter of a stretched wire) is 

crA = apjE. 

We shall see below that Poisson's ratio cannot exceed \. 
For most materials its value is in the range from 0-25 to 0-5. 
The value a = is reached in porous materials (such as cork) 
whose transverse dimensions are unaffected by stretching. 

Thus the elastic properties of a solid are described by two 
quantities, E and o\ It should be emphasised, however, that we 



§101] EXTENSION 297 

have tacitly assumed the solid to be isotropic (the materials 
concerned are usually polycrystalline). The deformation of an 
anisotropic body (a single crystal) depends not only on the 
position of the external forces with respect to the body but also 
on the position of the crystallographic axes within the body. The 
elastic properties of crystals are of course described by a larger 
number of quantities than for isotropic bodies. The number 
increases with decreasing symmetry of the crystal, from 3 for 
cubic crystals to 21 for crystals of the triclinic system. 

The work done on a body undergoing deformation is stored in 
the body in the form of elastic energy. Let us calculate this 
energy for a stretched rod. The work done by the tensile force 
F to increase the length of the rod by an infinitesimal amount 
d(l \) = lodX is 

dU = Fl d\, 

and this is also the increment of elastic energy. Substituting 
F = Sp, p = EX, and noting that the product Sl is the volume V 
of the rod, we obtain 

SEk.l d\ = VEkdk = VEd$\ 2 ). 

Hence it follows that, if the relative elongation of the rod changes 
from zero to some value A., the work done is ?VE\ 2 . Thus the 
elastic energy per unit volume of the deformed rod is 

U = $E\ 2 , 

which is proportional to the square of the deformation. This can 
also be put in the form 

U = $\p = p*l2E. 

An extension is a uniform deformation, i.e. one in which each 
volume element in the body is deformed in the same way. The 
bending of a thin rod is closely related to a simple extension 
(or compression), but is not a uniform deformation. Its nature 
is easily ascertained by imagining a rod bent into a circle. Before 
being bent the rod is straight, and so the length of each "fibre" 



298 



MECHANICAL PROPERTIES OF SOLIDS 



[XIII 



in it from one end to the other is the same. After the bending 
this is no longer true. The length of each fibre is 2nr, where r is 
the radius of the circle which it forms, and this radius is less along 
the inner side of the rod than along the outer side. It is therefore 
clear that the inner part of the rod is compressed and the outer 
part is stretched. Since no lateral force is applied to the rod 
surface, the elastic stresses in the rod act only lengthwise, and 
this means that, in bending, each volume element is subjected 
to a simple extension or compression, though this is not the same 
for different elements: the parts nearer to the convex side of the 
bent rod are stretched, and those nearer the concave side are 
compressed. 

§102. Uniform compression 

The formulae for a simple extension are easily generalised 
to any uniform deformations. 

Let a solid block in the form of a rectangular parallelepiped 
be stretched (or compressed) by forces acting on all sides and 
uniformly distributed over each face (Fig. 119). These forces 




Fig. 119. 



create elastic stresses in the body, which are in general different 
in three mutually perpendicular directions (along the three edges 
of the parallelepiped); let these stresses be p x , p y , p z , with 
positive signs for tensile stresses and negative signs for compres- 
sive stresses. The relative changes in length in these directions 
(positive in extension and negative in compression) will be 
denoted by \ x , \ y , k z . 



§102] UNIFORM COMPRESSION 299 

Let us consider this deformation as the result of three succes- 
sive simple extensions along the three axes. For example, when 
stretched by the stress p x the body is elongated in the x direction 
and shortened in the transverse y and z directions, with 

k x = pJE, k y =k z = —ak x = -apJE. 

Summation of the results of three such deformations gives 

= P x -(T(p y + p z ) ^ Py-O-JPx+Pz) _ Pz-<T(p x +Py) 



Next, let us find the change in the volume of the body as a 
result of the deformation. The volume of a parallelepiped with 
edges l x , l y , l z isV = l x l y l z . Taking logarithms, we find 

\0geV= \0%elx + log e / y + log e 4 

and on differentiating 

W^dl^ 8ly 8lz 

V III' 

V l x ly l Z 

The three terms in this sum are the relative elongations along the 
respective axes. Hence 

8V/V=\ x + ky+X z , 

i.e. the relative change in volume is equal to the sum of the relative 
elongations in three mutually perpendicular directions. 
Substitution of the expressions found above for k x , k y , k z gives 

8V l-2o- 

~y = E (Px+Py+Pz)- 

Let us now consider some important particular cases of uniform 
deformation. If a body is subject to tensile (or compressive) 
stresses which are uniform in all directions, i.e. if the elastic 
stresses in it are the same in all directions (p x = p y = p z ), then 
the relative change in each dimension of the body is the same 



300 MECHANICAL PROPERTIES OF SOLIDS [XHI 

(k x = X y = X g = a). Such a deformation is called a uniform 
extension (or compression); in this case 

\ = {\-2a)plE, 
and the relative change in volume is 

8V/V = 3X = plK, 
where the coefficient 

K = EI3(l-2a) 

is called the modulus of uniform compression or bulk modulus. 
Its reciprocal l/K is clearly equal to the compressibility 



1 



dV 



dp 



discussed in §58. Thus the formula obtained relates the ordinary 
compressibility of a solid to the values of Young's modulus and 
Poisson's ratio. 

The elastic energy stored in the body (per unit volume) in 
uniform compression is 

u = HKPx + Kpv + Kpz) = lAp = i/a 2 = hp 2 \K. 

The quantity K must always be positive, i.e. the volume of a 
body must be increased by extension and decreased by compres- 
sion. It has been mentioned in §70 that bodies with the opposite 
dependence of the volume on the pressure would be absolutely 
unstable and therefore cannot exist in Nature. [This is also seen 
from the above formula for the elastic energy: if K < 0, this 
energy would be negative, and since a mechanical system tends 
towards the state of least potential energy, such a body would 
spontaneously undergo an unlimited deformation.] 
Since K is positive, it follows that 1 — 2cr > 0, or 

o-<h 
i.e. Poisson's ratio cannot exceed h 



§103] SHEAR 301 

Let us now consider the compression of a block held by lateral 
walls in such a way that its transverse dimensions may be 
regarded as constant (Fig. 120); this process is called unilateral 
compression. 




Fig. 120. 



Let the direction of compression be along the x axis. The reac- 
tion of the walls which prevents a lateral expansion of the block 
gives rise to transverse stresses p y and p z in it. The magnitude of 
these is determined by the condition that the dimensions of 
the block in the y and z directions must remain unchanged 
(k y = k z = 0), and from symmetry we must have p y = p z . From 
the equation 



_ p y -o-JPx + Pz) _ P»(\ -a)- a-Px _ 



= 



we find that the transverse stresses are related to the compressive 
pressure p x by 



Pz = 



l-o- 



Px- 



The longitudinal compression of the block is given by 



v _ Px-<t(Pu+Pz) _ l-q--2o- 2 
kx ~ E E(l-a) Pa 



§103. Shear 

Under uniform compression, the shape of a body remains the 
same, and only its volume changes. Deformations of the opposite 
kind are also of importance, where only the shape of the body 
changes and not its volume. These are described as shear 
deformations. 



302 MECHANICAL PROPERTIES OF SOLIDS [XIII 

Since the volume is constant, we have 

8VIV=k x + k v + \ lc = 0, 
and hence 

Px + Pv + Pz = 0- 

Substituting p y +p z = —p x in the formula 

x = Px-<r(Pu + Pz) 
Ax E 

we find that the relative elongation (or shortening) along any 
edge of a block is related to the stress in that direction by the 
formula 

^x — 17 Px- 

This relation involves the quantity £7(1 + cr); a quantity equal to 
one-half of this is called the shear modulus (or modulus of 
rigidity) and denoted by G : 

G = E/2(l+o-). 

A shear deformation is, however, most simply brought about 
by applying to the block forces which are tangential, not perpen- 
dicular, to its surface. Let the lower face of the block be held 
fixed, and forces be applied in the plane of the upper face; 
stresses in this direction are often called shearing stresses. 
Under the action of these forces the parallelepiped becomes 
oblique, as shown in Fig. 121. The angle /3 (called the angle of 
shear) is small for small deformations (the only ones considered 
here). In a first approximation we can assume that the height of 
the parallelepiped is unchanged, and therefore that the volume 
is unchanged, giving a shear deformation. It can be shown that 
the angle /3 is related to the shearing force p (per unit area) by 

P = plG. 



§103] SHEAR 303 




y////////////////////////, 

Fig. 121. 

Like the modulus of uniform compression, the shear modulus 
must be positive, since the elastic energy that is stored in a body 
subjected to a shear deformation is positive only in that case. 
Hence it follows that we must have 1 + o- > 0, i.e. <x > — 1 . 

Using also the inequality a < \ derived in §102, we can say 
that the values of Poisson's ratio for all bodies must lie in the 
range 

-1 < o- < i 

These are the only conditions which follow from the general 
requirements of mechanical stability of solids. Thus in principle 
bodies could exist with negative values of o\ A rod of such a 
material should become wider in a simple extension, and not 
narrower as was assumed in §101. No bodies having such 
properties are known to exist in Nature, however, so that 
Poisson's factor in practice varies only between and \. Values 
close to \ occur in substances such as rubber, which change 
their shape considerably more easily than their volume: their 
moduli of compression are large in comparison with their shear 
moduli. 

The shearing of a rectangular block discussed above is a uni- 
form deformation. The torsion of a rod is a pure shear but one 
which is not uniform. This occurs when one end of a rod is fixed 
and the other end is twisted. Different cross-sections of the rod 
are turned through different angles relative to the fixed base. 
Since neither the height nor the cross-sectional area of the rod is 
changed, its volume also remains constant. 

It is easy to see how the shear deformation in torsion is 
distributed over the volume of the rod. Let us consider a rod of 
circular cross-section with radius R, and let its upper end turn 
through some angle <j> relative to the lower end (Fig. 122). Any 



304 



MECHANICAL PROPERTIES OF SOLIDS 



[XIII 




Fig. 122. 



generator AB of the cylindrical surface of the rod then moves to 
the oblique position AB'. Since the distance BB' = /?</>, the small 
angle of shear f3 on the surface of the rod is 

« tan /3 = R(f>ll, 

where / is the length of the rod. Applying the same reasoning to 
a cylindrical surface of radius r < R, we find that it is likewise 
sheared, but through an angle 

/3 r = r0//, 

which is less than the angle of shear (3 at the surface of the rod. 
Thus in torsion the different elements of the rod undergo different 
degrees of shearing, which become smaller as the axis of the rod 
is approached. 

The deformation in a twisted rod gives rise to elastic forces 
which counterbalance the applied forces. Since the elements 
of the rod can turn about its axis, the equilibrium condition is, 
as we know from mechanics, that the elastic and applied torques 
are equal. Hence it follows that the magnitude of the torsional 
deformation must be determined by the applied torque about the 
rod axis (also called the torsional torque). For small deforma- 
tions (when the angle of shear f3 is small), Hooke's law is valid 
and the angle of twist of the rod is proportional to the torsional 
torque. 



§104] PLASTICITY 305 

The relation between the angle of twist and the torsional torque 
can be used to measure the latter. This method of measuring 
torques is widely used in physics in what is called a torsion 
balance. Here the "rod" usually consists of a fine quartz thread 
of thickness from 1 to 100 /x, having high sensitivity and strength; 
the angle of twist of the thread is measured from the movement of 
a light spot reflected from a mirror fixed to the thread. Extremely 
small torques can be measured by means of such a balance. An 
intrinsic limit of sensitivity is imposed only by the spontaneous 
random vibrations of the balance due to unavoidable thermal 
fluctuations (similar to Brownian motion). As an example, the 
amplitude of the fluctuation torsional oscillations of a balance 
with a quartz thread 10 cm long and 1 n thick is only a fraction 
of a minute of arc at room temperature. 

§104. Plasticity 

There is a fundamental difference between compression (or 
extension) and shear deformations, which may be explained 
as follows. Let us consider a body undergoing shear, for example 
a cube of some material placed in a rigid container in the form 
of an oblique parallelepiped of equal volume. As a result of the 
shear, the body will contain some stored elastic energy. 

It is easily seen that the configuration of the atoms in the de- 
formed cube is not energetically advantageous. In other words, 
their configuration does not correspond to stable equilibrium 
(for a given shape of the body). For let us imagine that the 
container is filled with the material of the cube in molten form. 
By allowing this to solidify we obtain a body for which the shape 
of the container is natural and the shape of the cube is unnatural. 
The new configuration of the atoms is evidently one of lower 
energy, since it does not possess the shear energy. 

We see that a shear deformation is essentially unstable, since 
the atoms can be arranged within the boundaries of the deformed 
body in such a way that the energy of the body is decreased. 

This conclusion clearly applies only to shear and not to uni- 
form compression. Under compression, the elastic energy results 
from the change in volume of the body, and therefore can not be 
eliminated by any movement of the atoms within a fixed volume. 

If a shear deformation of a body were to be accompanied by a 
change in the configuration of atoms such as to eliminate the 



306 MECHANICAL PROPERTIES OF SOLIDS [XIII 

elastic energy, then the body would retain its different shape 
when the external forces were removed, and would not revert 
to its original shape. Such deformations which remain when the 
external forces cease to act are called plastic deformations. 

It is found that plastic deformations do not occur for stresses 
below a certain value, and the deformation disappears when the 
external forces cease to act. Such deformations are said to be 
elastic, and the whole of the discussion in §§101-3 refers only 
to these. 

The value of the stress for any given body above which a 
plastic deformation occurs is called the elastic limit. For smaller 
stresses, the body returns to its original state when the load is 
removed; for larger stresses, residual plastic deformations 
remain in the body after removal of the load. 

The value of the elastic limit depends not only on the substance 
but also to a considerable extent on the mode of preparation of 
the sample, its previous treatment, the presence of impurities 
etc. For example, the elastic limit of single crystals of aluminium 
is only about 4 kgf/cm 2 , but that of commercial aluminium is 
1000kgf/cm 2 . The elastic limit of heat-treated carbon steel 
reaches 6500 kgf/cm 2 . 

The elastic limit is very small in comparison with the shear 
modulus, and the limiting value of the deformation beyond which 
plasticity occurs is in general very small. For example, the shear 
modulus of aluminium is 2-5 X 10 5 kgf/cm 2 . This means, for 
example, that single crystals of aluminium are elastic only up to 
relative deformations A = 4/(2-5 X 10 5 ) ~ 10~ 5 . Steel is elastic 
up to \ ~ 10~ 2 . 

Plastic deformation itself affects the elastic limit of a body: 
when a body undergoes a plastic deformation, its elastic limit 
is raised. This is called hardening. For example, the elastic 
limit of a single crystal of zinc is so small that it can easily 
be bent with the fingers, but it is not so easily straightened 
again, since the bending increases the elastic limit. The phenome- 
non of hardening is, in particular, the basis of the change in 
properties of a metal by the process of cold working, which 
consists in plastically deforming it in some way. 

Owing to hardening, a body subject to stresses which exceed 
the elastic limit does not break. It undergoes a plastic deformation 
which increases until the resulting changes cause the elastic 



§104] PLASTICITY 307 

limit to become equal to the stresses caused by the external 
forces. We may say that the elastic limit is equal to the stress 
which caused the last preceding plastic deformation of the body. 
Figure 123 shows a diagram of the relation between the stresses 
p acting in the body and the magnitude k of the deformation. 




Fig. 123. 



If the stress is less than the elastic limit p , the deformation is 
elastic and obeys Hooke's law (more or less), i.e. k is pro- 
portional to p. This relation is shown in the diagram by the 
straight line OA. 

When the stress becomes greater than p , a plastic deforma- 
tion of the body occurs, and as the stress increases the relation 
between k and p is as shown by the curve AB, Let us suppose 
that, having reached a points' on this curve, we then decrease p. 
The value of p = p ' corresponding to A' is also the elastic 
limit acquired by the body through hardening as the load is 
increased. Thus, when p decreases, there will be no further 
plastic deformation, and the variation of k is shown by the 
straight line A'O', which is parallel to the elastic part AO of the 
line OB. When the stress becomes zero, there remains some 
deformation X pl , which is a plastic deformation. The total de- 
formation at the point A' can be written as the sum of plastic 
and elastic parts, k pl = OO' and \ e i = O'a. 

If the stress is again increased, the same straight line O'A' is 
traversed until the value p ' is reached. On passing the threshold 
p ' we move from the line O' A' to curve A' B and the plastic 
deformation is increased, the elastic limit being thereby further 
raised. 

With increasing plastic deformation the elastic limit does not, 
however, increase indefinitely: there is a maximum value of the 



308 MECHANICAL PROPERTIES OF SOLIDS [XIII 

elastic limit which cannot be exceeded. This is called the yield 
point. Under a stress of this amount, the deformation of the body 
increases continuously and it begins to flow like a liquid. By 
applying high pressures it is possible, for example, to cause a metal 
to flow in a jet from an aperture in the cylinder of a hydraulic press. 

It is clear that stresses exceeding the yield point can never 
occur in a body for any deformation (except, of course, in uniform 
compression). 

The yield point may sometimes not be reached, since the 
body may fracture much sooner. In order to observe the yield 
phenomenon, it is best to use such deformations as unilateral 
compression or torsion. Simple stretching, on the other hand, 
easily causes fracture. 

The presence of small, frequently microscopic, cracks in a 
body plays an important part in fracture. These cracks may be 
either on the surface of the body or within it, for example slight 
gaps between the grains of a polycrystalline body. Such cracks 
act as levers to cause a considerable concentration of the external 
forces applied to the body, it is comparatively easy for the elastic 
stresses at the sharp end of a crack to reach values sufficient for 
further rupture of atomic bonds and lengthening of the crack, 
ultimately leading to complete fracture of the body. The impor- 
tance of the state of the surface of the body with regard to fracture 
is clearly shown by an experiment with a rock-salt crystal: if the 
crystal is immersed in water, the salt dissolves from its surface, the 
cracks present on the surface are eliminated, and the crystal under 
water is considerably more difficult to break than a crystal in air. 

Plastic deformation near the ends of cracks may blunt their 
points and thus decrease to some extent the concentration of 
elastic stresses near them. In this sense plasticity aids the 
resistance of a body to fracture, as is shown by the temperature 
dependence of brittleness in metals. For example, steel, which is 
difficult to break at ordinary temperatures, becomes brittle at 
low temperatures. This effect is largely due to decreasing plasticity 
at low temperatures, which will be further discussed in § 106. 

§105. Defects in crystals 

The very fact that the plastic properties of a body depend 
considerably on its previous treatment, the presence of impurities, 
etc., indicates that these properties are closely related to features 



§105] 



DEFECTS IN CRYSTALS 



309 



of the crystal structure of actual bodies which distinguish them 
from ideal bodies. 

Departures from the ideal crystal structure are called defects. 
The simplest type, which may be called point defects, consist in 
the absence of an atom from a lattice point (a free vacancy) or 
the replacement of the "correct" atom at a lattice point by a 
different (impurity) atom, the entry of an extra atom between 
lattice points, and so on. The departure from the regular structure 
of the lattice extends over a distance of the order of several 
lattice periods around such a point. 

The most important defects as regards the mechanical proper- 
ties of solids are, however, of another kind, which may be called 
line defects, since the departure from the regular structure of the 
lattice is concentrated near certain lines. These are dislocations. 

The dislocation shown in Fig. 1 24 may be regarded as a lattice 
defect caused by the presence in the lattice of an extra crystal 
half-plane inserted between two "regular" planes (layers of 




Fig. 124. 



atoms). The line of the dislocation (which in this case is called 
an edge dislocation) is a straight line perpendicular to the plane 
of the diagram, shown by the symbol _L; the "extra" layer of 
atoms lies above this symbol. The dislocation may also be 



310 



MECHANICAL PROPERTIES OF SOLIDS 



[XIII 



regarded as the result of displacing the upper part of the crystal 
shown diagrammatically in Fig. 125a by one lattice period (Fig. 
125b). 




Fig. 125. 

Another type of dislocation may be visualised as the result 
of "cutting" the lattice along a half-plane and then displacing 
the parts of the lattice on either side of the cut by one lattice 
period along the edge of the cut (Fig. 126). The edge of the cut 
is then called a screw dislocation and is shown by the broken 
line in Fig. 126. The presence of such a dislocation converts 




Fig. 126. 



§105] DEFECTS IN CRYSTALS 311 

the crystal plane in the lattice into a helicoidal surface, like a 
spiral staircase without the steps. 

In an edge dislocation the displacement is perpendicular to 
the dislocation line, but in a screw dislocation it is parallel to 
\this line. Any intermediate case between these two extremes is 
possible. The dislocation lines need not be straight: they may be 
curves or even closed loops. 

There are various methods of directly observing dislocations. 
For example, in transparent crystals this can be done by creating 
supersaturated solid solutions of certain substances. The impurity 
atoms tend to be deposited as colloidal particles, which grow 
mainly at the places where the basic lattice structure is per- 
turbed; thus the colloidal particles of impurities are concentrated 
along dislocation lines and render them visible. Another method 
is based on the etching of the crystal surface by suitable reagents. 
The surface is more easily attacked at points where the crystal 
structure is perturbed. This leads to the formation of visible pits 
at points where the dislocation lines reach the surface of the 
crystal. 

Screw dislocations often play a decisive part in the process of 
growth of crystals from a liquid or a supersaturated vapour. 

It has been shown in §99 how the formation of a new phase 
within the original phase must begin with nucleation. A similar 
situation must occur in the growth of a crystal. The formation 
of a new layer of atoms on a perfectly regular crystal surface 
cannot begin simply with the deposition of individual atoms on 
the surface: such atoms, having neighbours on one side only, 
would be under conditions which would be energetically very 
unfavourable, and would not remain on the surface. A stable 
"nucleus" for a new layer of atoms on the surface of the crystal 
must contain immediately a sufficient number of atoms, and the 
chance occurrence of such nuclei may be comparatively rare. If, 
however, the edge of a screw dislocation appears on the surface 
of the crystal, it provides a ready-made step (one atomic layer 
in height) to which new atoms can easily attach themselves, and 
no nuclei are therefore necessary. The rate of attachment of new 
atoms is approximately the same all along the edge of the step. 
The crystal therefore grows spirally, as shown diagrammatically 
in Figs. 127a-d. At any time there is a free step on the surface 
of the crystal, which can therefore grow without limit. The rate 



312 



MECHANICAL PROPERTIES OF SOLIDS 



[XIII 




(a) 



(b) 




(c) 




Fig. 127. 



of growth is very much higher than that of a process depending 
on nucleation. 



§106. The nature of plasticity 

Groups of parallel lines can often be observed on the surface 
of a single crystal undergoing a plastic shear deformation. These 
lines are the traces of the intersection of the surface of the body 
with the slip planes along which some parts of the crystal slide 
as a whole relative to other adjoining parts. Thus the plastic 
deformation is non-uniform: large displacements in shear occur 
only along planes at a comparatively large distance apart, while 
the parts of the crystal which lie between these planes undergo 
almost no deformation. Fig. 128 is a diagram of the deformation 
of a body as a result of slipping of this kind. 

The configuration of the slip planes is closely related to the 
structure of the crystal lattice. In any crystal, slipping occurs 
almost entirely along certain planes; for example, in the NaCl 
crystal these are (110) planes, while in metal crystals with face- 
centred cubic lattices they are (111) planes. 



§106] THE NATURE OF PLASTICITY 313 

What is the mechanism whereby one part of a crystal slips 
relative to another? If this were to take place simultaneously over 
the whole slip plane, very large stresses would be necessary. 




Fig. 128. 

The change from one equilibrium configuration of atoms to 
another (say, from that shown in Fig. 125a to that in Fig. 125d) 
would have to occur by means of a large elastic deformation in 
which the relative displacements (in the region near the slip 
plane) would reach values A ~ 1. This would require stresses 
of the order of the shear modulus G. 

In reality, the elastic limits of actual bodies are usually 10 2 
to 10 4 times less than their shear moduli, and so relatively small 
stresses are necessary in effecting a shear. This is possible 
because slip actually takes place by the movement of dislocations 
in crystals. 

The simplest form of this mechanism is indicated by the 
sequence in Figs. 125 a-d. If the crystal contains an edge disloca- 
tion (passing through a point A and at right angles to the front 
face of the crystal), the movement of this dislocation in the slip 
plane from left to right through the body causes a displacement 
of the upper part of the crystal relative to the lower part by one 
lattice period. The movement of the dislocation involves only 
a relatively slight reconstruction of the lattice, which affects only 
the atoms near a single line. This process may be compared to 
the movement of a wrinkle in a carpet: the wrinkle moves more 
easily than the whole carpet, but the effect of moving the wrinkle 
from one end of the carpet to the other is to shift the whole 
carpet a certain distance. 

Thus the plasticity of a solid depends on the presence of 
dislocations in it and on the possibility of their free movement. 



314 MECHANICAL PROPERTIES OF SOLIDS [XIII 

This movement may, however, be retarded by various obstacles, 
such as impurity atoms dissolved in the lattice or small solid 
inclusions in the body. Dislocations are also slowed down by 
intersecting one another, and by the grain boundaries in a poly- 
crystalline body. At the same time, the interaction of dislocations 
with one another and with other defects gives rise to new disloca- 
tions. These processes are very important, since they support 
the development of a plastic deformation; otherwise, the defor- 
mation would cease as soon as all the dislocations existing in 
the body had been "utilised". 

The number of dislocations in a body is described by the 
dislocation density, which is the number of dislocation lines 
intersecting a unit area within the body. This number varies 
widely, from 10 2 — 10 3 cm~ 2 in the most perfect pure single 
crystals to 10 11 — 10 12 cm~ 2 in heavily deformed (cold-worked) 
metals. 

It is clear from the above discussion that pure single crystals 
will have the lowest strength (i.e. the lowest elastic limit), since 
the dislocation density in them is comparatively low, and so there 
is practically no interference between the dislocations in their 
motion. Hardening of the material can be achieved by dissolving 
impurities in it or depositing microscopic solid inclusions, or 
by reducing the grain size. For example, the strength of iron is 
increased (in various kinds of steel) by dissolving in it carbon 
atoms or microscopic inclusions of iron carbide which are 
deposited in the process of solidification. 

Plastic deformation itself damages the crystal lattice, in- 
creasing the number of defects in crystals and thereby imped- 
ing the further movement of dislocations. This is the reason for 
the phenomenon of hardening by deformation, including the 
hardening of metals by cold working (work-hardening). 

The hardening achieved by plastic deformation is not main- 
tained for an indefinite time, however. The most stable state of 
a body is the undisturbed ideal crystal, which is the state having 
the least energy. Thus perturbed crystals exhibit what is called 
recrystallisation. The structural defects are "healed" and the 
large grains in a polycrystalline body increase in size at the 
expense of the smaller ones, resulting in a system which is less 
defective and therefore of lower strength. Recrystallisation 
occurs more rapidly at high temperatures, and especially rapidly 



§106] THE NATURE OF PLASTICITY 315 

at temperatures fairly near the melting point (for example, in 
the annealing of metals). At low temperatures there is practically 
no recrystallisation. The effect of recrystallisation is gradually 
to eliminate the hardening, and if the body is subject to a steady 
load it will slowly flow. 

The temperature also has a marked effect on the movement 
of dislocations. Since this movement involves the overcoming 
of potential barriers by the atoms (which change their con- 
figuration near the moving dislocation line), it is a process of the 
activation type (cf. §91), and therefore is rapidly stopped by 
lowering the temperature, thus decreasing the plasticity of the 
body. 

The methods described above for increasing the strength of 
a material are based on the creation of obstacles to the move- 
ment of dislocations. The opposite means of hardening is also 
possible, namely to produce a single crystal which contains 
no dislocations at all. Such a crystal should in principle have the 
maximum possible elastic limit: its plastic deformation could be 
brought about only by simultaneous slipping along entire planes, 
which, as already mentioned, would require the application of 
extremely large stresses. 

This ideal state is approached by what are called whiskers. 
These are extremely thin thread-like crystals with thicknesses of 
the order of microns. They are formed both by metals and by non- 
metals, and can be obtained in various ways: by precipitation of 
slightly supersaturated vapours of pure metals at appropriate tem- 
peratures in an inert gas medium, by slow precipitation of salts from 
solutions, and so on. In many cases such crystals appear to grow 
round individual screw dislocations in the manner described in 
§105. The dislocation along the axis of the whisker does not affect 
its mechanical properties when it is stretched, and the crystal 
behaves practically as an ideal one. 

It is clear from the above discussion that all these plasticity 
properties relate only to crystalline bodies. Amorphous bodies, 
such as glass, are not able to undergo plastic deformation, and are 
said to be brittle. Their inelastic behaviour consists of either 
fracture or a slow flow under the prolonged action of forces, in 
accordance with the fact that amorphous bodies are actually 
liquids of very high viscosity. 



316 MECHANICAL PROPERTIES OF SOLIDS [XIII 

§107. Friction of solids 

The sliding of a solid body on the surface of another body is 
always accompanied by the conversion of its kinetic energy into 
heat, and in consequence the motion is gradually retarded. This 
phenomenon can be described from the purely mechanical point 
of view as being due to a certain force which impedes the mot- 
ion, called a frictional force. Physically, friction is the result of 
complex processes which occur on surfaces which rub together. 

Experiment shows that the friction between solid bodies usually 
obeys certain simple laws. It is found that the total frictional 
force F fr acting between moving bodies is proportional to the force 
N which presses the bodies together, and does not depend on the 
area of contact between the bodies or on the speed of the motion: 

F fr = ^N. 

The quantity fi is called the coefficient of friction; it depends only 
on the properties of the surfaces which rub together. This rela- 
tion is usually satisfied to a good approximation over a wide 
range of experimental conditions (loads and rates of sliding), but 
deviations from it are also found. 

Friction depends considerably on the way in which the rubbing 
surfaces have been treated and on their present state (whether 
contaminated, and by what). For example, the coefficient of 
friction between metal surfaces is usually in the range from 0-5 to 
1-5. These values, however, are for metal surfaces exposed to 
the air. Such surfaces are always contaminated by oxides, ad- 
sorbed gases, etc., which impair the conditions of contact. Ex- 
periment shows that completely clean metal surfaces prepared 
by heating in vacuum show very high friction in sliding, and 
sometimes "stick" completely. 

There is probably no single universal mechanism of friction, 
and the nature of friction is dilferent for surfaces of different 
types and with different previous treatment. As an illustration we 
shall describe the mechanism of friction for certain metals. 

Experiment shows that metal surfaces always exhibit irregu- 
larities which are large in comparison with molecular distances. 
Even for surfaces prepared and polished in the best possible way, 
the depth of the irregularities is 100-1000 A, and rubbing sur- 
faces in engineering usually have much greater non-uniformities. 



§107] FRICTION OF SOLIDS 317 

When bodies touch, the actual contact between them occurs only 
at the "peaks" of these non-uniformities. Thus the area of actual 
contact S may be very small in comparison with the total nomi- 
nal area of contact S; the ratio may be 10" 4 or 10~ 5 . In plastic 
metals, even under small loads, the "peaks" of the non-unifor- 
mities are deformed and flattened until the true pressure acting on 
them decreases to a certain value p lim below which the deforma- 
tion ceases. The area of contact 5 is determined by the condition 
PumSo = N, and is therefore proportional to the load N. In the 
regions of actual contact, the forces of molecular cohesion bring 
about a strong "adhesion" of the bodies. During sliding there is 
a continual separation and formation of fresh regions of contacts. 
The force required to break contact is proportional to the area of 
contact S , and therefore to the load N. 

The frictional force during motion must be distinguished from 
the force needed at the beginning of the motion in order to start 
the body from rest. This limiting friction is also proportional to 
the load, but the coefficient is somewhat greater than in motion 
(although the difference is not more than 10-20%). 

It should be emphasised that the whole of the above discussion 
refers to friction between dry surfaces of solid bodies. It bears no 
relation to the friction between lubricated surfaces separated by a 
layer of fluid. In the latter case the frictional force is due to the 
viscosity of the liquid; a simple example of this type of friction 
will be discussed in § 1 19. 

As well as sliding friction, there is also the friction that occurs 
when one body rolls on another. Let us consider a cylinder of 
radius r rolling on a plane. In order to overcome the frictional 
force and maintain steady rolling, a force F must be applied, 
which is described by the torque K about the instantaneous line 
of contact between the cylinder and the plane; if the force is 
applied to the axis of the cylinder, then K = rF. The torque K is 
a measure of the rolling friction; it is found to be proportional 
to the force N which presses the rolling body to the surface on 
which it rolls: 

K = yN. 

The coefficient y depends on the two bodies in contact; it clearly 
has the dimensions of length. 



CHAPTER XIV 



DIFFUSION AND THERMAL 
CONDUCTION 



§108. The diffusion coefficient 

In the preceding chapters we have discussed mainly the 
properties of bodies in thermal equilibrium. This chapter and 
the next deal with processes by means of which a state of 
equilibrium is reached, called kinetic processes. These are all 
essentially irreversible processes, since they bring a body closer 
to equilibrium. 

If a solution has different concentrations at different points, 
the thermal motion of the molecules causes mixing of the solution 
in the course of time: the solute moves from regions of higher to 
regions of lower concentration, until the composition of the 
solution becomes uniform throughout its volume. This process is 
called diffusion. 

For simplicity, let us assume that the concentration of the 
solution (denoted by c) varies only in one direction, which we 
shall take as that of the x axis. The diffusion flux j is defined as 
the quantity of solute passing per unit time through a surface of 
unit area perpendicular to the x axis, and will be taken as positive 
if the flux is in the positive direction of this axis, and negative if 
it is the opposite direction. Since matter passes from regions 
of higher to regions of lower concentration, the sign of the flux 
is opposite to that of the derivative dc/dx (called the concentra- 
tion gradient): if the concentration increases from left to right, 
the flux is to the left, and conversely. If dc/dx — 0, i.e. the con- 
centration of the solution is constant, there is no diffusion flux. 

All these properties are included in the following relation 
between the diffusion flux and the concentration gradient: 

j=—D dc/dx. 
318 



§109] THE THERMAL CONDUCTIVITY 319 

Here D is a constant coefficient called the diffusion coefficient. 
This relation describes the properties of diffusion "phenomeno- 
logically", that is, from its external manifestations. We shall 
see below (§113) how a similar expression for the flux can be 
derived directly by considering the molecular mechanism of 
diffusion. 

The flux j in the above formula may be defined in any manner: 
as the mass of solute passing through unit area, as the number of 
solute molecules, and so on, but the concentration c must then 
be defined in a similar manner as the mass or number of molecules 
of solute per unit volume. Then it is evident that the diffusion 
coefficient will not depend on the way in which the flux and the 
concentration are defined. 

The dimensions of the diffusion coefficient may be found as 
follows. Let j be the number of solute molecules passing through 
unit area per unit time. Then [/] = 1 /cm 2 . sec. The concentration 
is the number of solute molecules per unit volume, with dimen- 
sions [c] = 1/cm 3 . Comparing dimensions on the two sides of 
the equation j = — D dcjdx, we find 

[D] = cm 2 / sec. 

When speaking of diffusion, we imply that it occurs in a 
medium at rest, so that the equalising of the concentration occurs 
only because of the random thermal motion of the individual 
molecules. It is assumed that the liquid (or gas) is not mixed by 
any external interaction which causes it to move. 

Such mixing may occur in a liquid, however, because of 
gravity. If a light liquid such as alcohol is carefully poured on 
water, the liquids will mix by diffusion, but if water is poured 
on alcohol, streams of water (the heavier liquid) will descend and 
streams of alcohol will rise. 

Thus gravity may cause the composition of a medium to be 
equalised by movement. This is called convection; it equalises 
the concentration much more rapidly than diffusion. 

§ 1 09. The thermal conductivity 

The process of thermal conduction is akin to diffusion. If the 
temperature is different at different points in a body, a heat flux 
occurs from hotter to colder regions, and continues until the 



320 DIFFUSION AND THERMAL CONDUCTION [XIV 

temperature is the same throughout the body. Here again the 
mechanism of the process is based on the random thermal motion 
of the molecules: molecules belonging to the hotter parts of the 
body collide with molecules in adjoining colder parts and transmit 
to them part of their energy. 

As in the discussion of diffusion, it is assumed that thermal 
conduction takes place in a medium at rest. In particular, it is 
assumed that the medium contains no pressure variations which 
would cause motion in it. 

Let us suppose that the temperature T of the medium varies 
only in one direction, which we again take as that of the x axis. 
The heat flux q is defined as the quantity of heat passing per unit 
time through unit area perpendicular to the x axis. Just as for 
diffusion, the relation between the heat flux and the temperature 
gradient dTjdx is 

q = — k dTjdx. 

Here again the minus sign appears because the direction of the 
heat flux is opposite to that in which the temperature increases: 
heat flows in the direction of decreasing temperature. The 
coefficient k is called the thermal conductivity. 

If the quantity of heat is measured in ergs, the heat flux will 
be measured in erg/cm 2 . sec, and the dimensions of the thermal 
conductivity are therefore 

[k] = [erg/cm. sec. deg] 
= [g.cm/sec 3 .deg] . 

The thermal conductivity determines the rate of flow of heat 
from hotter to colder regions. The change in temperature of a 
body is equal to the quantity of heat gained, divided by the 
specific heat. Thus the rate of equalisation of the temperature at 
different points in the body is governed by the thermal conduc- 
tivity divided by the specific heat per unit volume, i.e. the 
quantity 

X = K/pc p , 

where p is the density and c p the specific heat per unit mass (at 
constant pressure, since thermal conduction at constant pressure 
is being discussed). This quantity is called the thermal diffusivity. 



§110] THERMAL RESISTANCE 321 

It is easily seen to have the dimensions 

[x] = cm 2 /sec, 

which are the same as those of the diffusion coefficient. This is 
natural, since if both sides of the relation q = —K dTldx are 
divided by pc p the ratio qlpc p on the left-hand side may be 
regarded as a "temperature flux", i.e. the flux of the quantity 
whose gradient appears on the right. Thus the coefficient x is a 
kind of diffusion coefficient for temperature. 

As with diffusion, the action of gravity may cause convective 
mixing of a non-uniformly heated liquid (or gas). This occurs 
when the liquid is heated below (or cooled above): the hotter 
and therefore less dense lower layers of the liquid rise and are 
replaced by descending currents of colder liquid. The equalisation 
of temperature by convection occurs, of course, much more 
rapidly than by thermal conduction. 

As examples, the following table shows the values of the ther- 
mal conductivity for a number of liquids and solids (at room 
temperature). These values are given in units of J/cm. sec. deg, 
i.e. the heat flux is defined as the energy in joules transported 
through 1 cm 2 in 1 sec. 

Water 6-OxKr 3 Lead 0-35 

Benzene 1-5 x 1(T 3 Iron 0-75 

Glass 4to8xl(T 3 Copper 3-8 

Ebonite l-7xl(r 3 Silver 4-2 

The very high thermal conductivity of metals should be noted. 
The reason for this is that in metals, unlike other bodies, heat is 
transferred by the thermal motion of free electrons, and not 
of atoms. The effectiveness of heat conduction by electrons is 
due to their high velocity, of the order of 10 8 cm/sec, which is 
much higher than the ordinary thermal velocities of atoms and 
molecules (10 4 to 10 5 cm/sec). 

§ 1 1 0. Thermal resistance 

The simple relation given above between the heat flux and the 
temperature gradient makes possible the solution of various 
problems relating to thermal conduction. 



322 DIFFUSION AND THERMAL CONDUCTION [XIV 

Let us consider a layer of material (of thickness d) between two 
parallel planes, each of area S, and assume that these boundary 
planes are maintained at different temperatures 7\ and T 2 (with 
T x > T 2 ). The thermal conductivity of the substance is in general 
a function of temperature, but we shall suppose that the difference 
between the temperatures T x and T 2 is not very great, so that we 
may neglect the variation in the conductivity across the thickness 
of the layer and regard k as a constant. 

Let the jc axis be taken across the thickness of the layer, and let 
x be measured from the plane at temperature 7\. It is evident that 
a temperature distribution depending only on x will be established 
in the layer of material, and a heat flux through the layer from 
7\ to T 2 will exist. Let us find the relation between this flux and 
the temperature difference 7\ — T 2 which causes it. 

The total heat flux Q through the whole cross-section of the 
layer (parallel to the boundary planes) per unit time is equal to the 
product qS of the flux q per unit area and the total area S of the 
cross-section. Using the relation between q and the temperature 
gradient, we can write 

Q = -kS dTldx. 

The flux Q is clearly independent of x, since no heat is absorbed 
in passing through the layer and none is evolved within the layer; 
the total quantity of heat passing per unit time through any sur- 
face which intersects the entire layer must therefore be the same. 
From the above equation we therefore have 

T = —(QIkS)x + constant, 

i.e. the temperature varies linearly across the thickness of the 
layer. When jc = 0, i.e. on one of the boundary planes, we must 
have T = IV, hence the constant is equal to 7\, and 

T=T 1 -(QIkS)x. 

At the other boundary plane (x = d) we must have T = T 2 , i.e. 

T 2 =T 1 -(Q/ K S)d. 



§110] THERMAL RESISTANCE 323 

Hence 

Q = {KSld)(T x -T 2 ). 

This formula gives the required relation between the heat flux Q 
and the temperature difference across the layer. 

Let us now consider a layer of material bounded by two con- 
centric spheres (of radii r x and r 2 ) maintained at temperatures T x 
and T 2 . Figure 129 shows a central cross-section. The tempera- 
ture at any point within the layer is evidently a function only of 
the distance r from the centre of the spheres. 




Fig. 129. 

Since the only coordinate on which the temperature depends in 
this case is r, the heat flux q is everywhere in the radial direction, 
and is 

q = —K dT/dr. 

The total heat flux through a spherical surface of radius r con- 
centric with both spheres and lying between them is 

Q = Airf-q = —Anicr 2 dT/dr, 
whence 

dTldr = -QI4irKr*. 



324 DIFFUSION AND THERMAL CONDUCTION [XIV 

As in the previous case, the total heat flux through any closed sur- 
face enclosing the inner sphere must be the same, and Q is 
therefore independent of r. The above equation then gives 

T = Y constant. 

47r/cr 

The constant is determined by the condition that T = 7\ for r = r u 
so that 

Finally, from the condition that T = T 2 for r = r 2 we obtain the 
following relation between the total heat flux and the temperature 
difference across the layer: 

n _ (T 1 -TJ.4nK 

In particular, if r 2 = », i.e. if there is an infinite medium round a 
spherical surface of radius r x (T 2 in this case being the tempera- 
ture at infinity), the expression for the heat flux becomes 

G = 4flricr 1 (r 1 -7' 8 ). 

The ratio of the temperature difference at the boundaries of a 
body to the total heat flux is called the thermal resistance of the 
body. The above formulae show that the thermal resistance of a 
plane slab is d/rcS, and that of a spherical layer is 

1 /l 1 



/!- 



4 7TK\r r r 2 ) 

Entirely similar results are evidently obtained for diffusion in a 
solution bounded by two planes or two spherical surfaces on 
which given concentrations are maintained. In the above for- 
mulae we need only replace the temperature by the concentration, 
the heat flux by the diffusion flux and k by the diffusion coefficient 
D. 



§110] THERMAL RESISTANCE 325 

Let us apply these formulae to the problem of rate of melting, 
and consider a piece of ice immersed in water at a temperature 
T x above 0°C. Since equilibrium between ice and water is 
possible (at atmospheric pressure) only at a definite tempera- 
ture r = 0°C, the water immediately adjoining the ice will be at 
this temperature. At increasing distances from the ice, the water 
temperature is greater and tends to T t . There will be a heat flux 
from the water to the ice. On reaching the ice, the heat is absorbed 
as the heat of fusion necessary to convert ice into water. For 
example, if the piece of ice is spherical (with radius r ), it will 
receive per unit time from the surrounding water (which we regard 
as an infinite medium) a quantity of heat 

G = 4ir#cr (r 1 -r«). 

Dividing this by the heat of fusion, we find the quantity of ice 
which melts per unit time. Thus the rate of melting is determined 
by the process of thermal conduction in the surrounding water. 

Similarly, the rate of dissolution of a solid in a liquid is deter- 
mined by the rate of diffusion of solute in the liquid. Near the 
surface of the solid, a thin layer of saturated solution is imme- 
diately formed; further dissolution takes place as the solute 
diffuses from this layer into the surrounding liquid. For example, 
if the solid is a sphere of radius /■„, the total diffusion flux J from 
the sphere into the solvent, which is the quantity of substance 
dissolving per unit time, is 

J = AttDtqCq. 

Here c is the concentration of the saturated solution, and the 
concentration in the liquid at a great distance from the sphere is 
taken to be zero. 

Processes of diffusion and thermal conduction also determine 
the rate of evaporation of a liquid drop in a gas of another sub- 
stance, such as air. The drop is surrounded by a layer of saturated 
vapour, from which the substance slowly diffuses into the sur- 
rounding air. The process of heat transfer from the air to the drop 
is also of importance. 

These examples are typical in that the rates of phase transitions 
occurring under steady-state conditions are usually determined 
by processes of diffusion and thermal conduction. 



326 DIFFUSION AND THERMAL CONDUCTION [XIV 

§111. The equalisation time 

If the concentration of a solution is different at different points, 
then, as we know, the composition will be equalised in the course 
of time by diffusion. Let us determine the order of magnitude of 
the time t required for this process. This may be done from 
considerations of the dimensions of the quantities on which this 
time can depend. 

First of all, it is evident that the time t cannot depend on the 
actual concentrations of the solution, for if all the concentrations 
are changed by a given factor, the diffusion flux which equalises 
the concentrations is changed by the same factor, and the 
equalisation time therefore remains unchanged. 

The only physical quantities on which the time t of diffusion 
equalisation can depend are the diffusion coefficient D in the 
medium concerned and the size of the region in which the con- 
centrations are different; let the linear size of this region be of 
order of magnitude L. 

The dimensions are [D] = cm 2 /sec, [L] = cm. It is evident that 
only one combination having the dimensions of time can be 
formed from these quantities, namely L 2 /D, and this must give 
the order of magnitude of the time t: 

t ~ L 2 ID. 

Thus the time for equalisation of concentrations in a region of 
size L is proportional to the square of L and inversely proportional 
to the diffusion coefficient. 

This question can be inversely stated as follows. Let us suppose 
that at some initial instant there is a certain quantity of solute 
concentrated in a small region of the solvent. In time this accumu- 
lation of solute will be dispersed by the effect of diffusion, and 
will be distributed throughout the whole large volume of the 
solvent. What is the mean distance L traversed by the diffusing 
substance in a time r? That is, we now wish to find the distance 
from the time, not the time from the distance. The answer is 
clearly given by the same formula, which must now be written 

l ~ \/(Dt). 

Thus in a time t the diffusing substance spreads to a distance 
proportional to V7 . 



§111] THE EQUALISATION TIME 327 

This relation may also be regarded in another way. Let us 
consider any one molecule of solute in the solution. Like all 
molecules it has a random thermal motion. We may ask what is 
the order of magnitude of the distance which this molecule can 
traverse from its initial position in time t; in other words, what 
is the mean straight-line distance between the initial and final 
positions of a molecule which has moved for a time t. Instead of 
considering a single molecule, let us suppose that there is a very 
large number of molecules close together. Then, as we have seen, 
in the course of time these molecules will move apart in all 
directions by diffusion, and the average distance travelled is 
L ~ \/{pi). This distance L is clearly also the mean distance that 
each molecule moves from its original position in time /. 

This result applies not only to molecules of solute but also to 
any particles suspended in a liquid and executing Brownian 
motion. 

The above discussion has referred entirely to diffusion, 
but the same arguments apply also to thermal conduction. We 
have seen in §109 that in the propagation of heat the diffusion 
coefficient is replaced by the thermal diffusivity x- Thus the 
temperature equalisation time in a body of linear size L is 

/ ~ L 2 / x ~ L 2 P cJk. 

This relation also can be inverted as was done above for the 
case of diffusion. In this connection let us consider the following 
problem. We assume that fluctuations of temperature with some 
frequency a> are artificially created on the surface of a body. These 
fluctuations will penetrate into the body, producing what is called 
a thermal wave. The amplitude of the fluctuations, however, will 
be damped with increasing depth in the body, and the question 
is to what depth L the fluctuations penetrate. Here the charac- 
teristic time is the period of the fluctuations, i.e. the reciprocal 
of the frequency. Substituting l/a> for t in the relation between the 
distance of heat propagation and the time, we obtain 

L ~ V( X /o>). 
This is the solution to the problem. 



328 DIFFUSION AND THERMAL CONDUCTION [XIV 

§112. The mean free path 

Turning now to discuss thermal conduction and diffusion 
in gases, we must first consider the nature of the interaction 
between gas molecules in somewhat more detail than hitherto. 

Gas molecules interact by means of collisions. During the 
greater part of the time, the molecules are comparatively far 
apart and move as if free, scarcely interacting at all. The mole- 
cules interact only during short intervals of time when they 
collide with one another. In this respect a gas differs from a 
liquid, in which the molecules are continuously interacting, and 
they cannot be said to undergo separate "collisions". 

Molecules may collide in various ways. Strictly speaking, in 
each passage of molecules at not too great a distance they 
undergo some change in velocity, and the concept of a "collision" 
is therefore not entirely precise. In order to make the concept 
more definite, we shall regard as collisions only those cases 
where the molecules pass so close that the interaction consider- 
ably alters their motion, i.e. their velocities are considerably 
changed in magnitude or direction. 

Collisions between molecules in a gas occur completely ran- 
domly, and the distance travelled by a molecule between two 
successive collisions may therefore have any value. We can, 
however, define a mean value of this distance, which is called the 
mean free path of the molecules, and is an important molecular- 
kinetic property of the gas; it will be denoted by /. As well as 
the mean free path, we may consider also the mean time t 
between two successive collisions. In order of magnitude, 
evidently, 

T ~ I/V, 

where v is the mean velocity of thermal motion of molecules. 
Let us consider two colliding molecules, regarding one of 
them as being at rest in a certain plane, and the other as crossing 
this plane. As explained above, the molecules will be said to 
collide only when they pass so close that their motion is con- 
siderably altered. This means that the moving molecule collides 
with the stationary one only if it meets the plane somewhere 
within a certain small region around the fixed molecule. This 
"target" area which the molecule must strike is called the effective 



§112] THE MEAN FREE PATH 329 

cross-section (or simply the cross-section) for collisions, and will 
be denoted by cr. 

As an example, let us determine the collision cross-section 
for molecules regarded as solid spheres of radius r . The greatest 
distance between the centres of two spheres at which they can 
pass and still touch is 2r . Thus the "target" area which the 
molecule must strike if a collision is to occur is a circle of radius 
2r round the centre of the stationary molecule. Thus the collision 
cross-section in this case is 

o- = 4tjt 2 , 

or four times the cross-sectional area of the sphere. 

In reality, of course, molecules are not solid spheres, but since 
the interaction force between two molecules decreases very 
rapidly with increasing distance between them, collisions occur 
only if the molecules almost "graze" each other. The collision 
cross-section is therefore of the order of magnitude of the cross- 
sectional area of the molecule. 

Let a molecule traverse a distance of unit length in its motion, 
and let us imagine the molecule as sweeping out a volume of unit 
length and cross-sectional area or; the magnitude of this volume 
is also cr. The molecule collides with all molecules lying within 
this cylinder. Let n be the number of molecules per unit volume. 
Then the number of molecules in the volume cr is no; and the 
molecule therefore undergoes no- collisions per unit length of 
path. The mean distance between two collisions, i.e. the mean 
free path, is in order of magnitude 

/ ~ 1/rtcr. 

It is seen from this expression that the mean free path is in- 
versely proportional to the gas density and depends on no other 
quantity. 

It must be remembered, however, that this last statement is 
valid only if the cross-section is assumed constant. Because 
the repulsion forces increase very rapidly as the molecules 
approach, molecules usually behave qualitatively as elastic 
solid particles, which interact only when they "graze" each other. 
Under these conditions the collision cross-section is in fact a 



330 DIFFUSION AND THERMAL CONDUCTION [XIV 

constant (depending only on the nature of the molecules). There 
are also, however, weak forces of attraction between molecules 
at greater distances. As the temperature decreases, the velocities 
of the gas molecules become less, and thus the duration of a 
collision between two molecules (passing at a given distance) 
increases. Because of this "lengthening" of the collision the 
motion of the molecules may be considerably changed even if 
they pass relatively far from each other. Thus, when the tempera- 
ture decreases, the collision cross-section increases somewhat. 
For example, in* nitrogen and oxygen cr increases by about 30% 
when the temperature falls from +100°C to — 100°C, and in 
hydrogen by 20%. 

For air at 0°C and atmospheric pressure, n ~ 3 x 10 19 . The 
cross-section o-«5x 10 -15 cm 2 , and therefore the mean free 
path of the molecules / «= 10~ 5 cm. The mean thermal velocity 
of the molecules t;«5x 10 4 cm/sec, and the time between 
collisions is accordingly t~2x 10~ 10 sec. The mean free path 
increases rapidly with decreasing pressure. For instance, at an 
air pressure of 1 mm Hg / ~ 10 -2 cm; in a high vacuum of the 
order of 10~ 6 mm Hg pressure, the mean free path reaches 
values of tens of metres. 

§113. Diffusion and thermal conduction in gases 

By means of the concept of the mean free path we can deter- 
mine the order of magnitude of the diffusion coefficient and the 
thermal conductivity and ascertain how they depend on the state 
of the gas. Let us take first the diffusion coefficient and consider 
a mixture of two gases whose total pressure is everywhere con- 
stant but whose composition varies in one direction, which we 
take as that of the x axis. 

Let n x be the number of molecules of one of the gases in the 
mixture per unit volume; this number is a function of the co- 
ordinate x. The diffusion flux j is the number of molecules pass- 
ing per unit time through unit area perpendicular to the x axis 
and moving in the positive direction of that axis, minus the 
corresponding number moving in the negative direction. 

The number of molecules passing through unit area per unit 
time is equal in order of magnitude to n x v , where v is the mean 
thermal velocity of the molecules. Here we may suppose that 
the number of molecules crossing this area from left to right 



§113] DIFFUSION AND THERMAL CONDUCTION IN GASES 331 

is determined by the density n t at the point where the molecules 
underwent their last collision, i.e. at a distance / to the left of 
the area; similarly, for molecules going from right to left we must 
take the value of n t at a distance / to the right of the area. If the 
coordinate of the area itself is x, the diffusion flux is given by 

j ~ vn t (x — /) — vn x {x + /). 

Since the mean free path / is a small quantity, the difference 
n t (x — /)— ritix + l) may here be replaced by — / dnjdx. Thus 

j - — vl drill dx. 

Comparison of this expression with the formula j = —D dnjdx 
shows that the diffusion coefficient in a gas is in order of magni- 
tude 

D ~ vl. 

The mean free path / « Una, where n is the total number of 
molecules of the two gases per unit volume. Thus D may also 
be written as 

D ~ v/na. 

Finally, the equation of state of an ideal gas shows that the 
number density of molecules in it is n = kT/p, so that 

D ~ vkTlpcr. 

The diffusion coefficient in a gas is therefore inversely propor- 
tional to its pressure (at a given temperature). Since the thermal 
velocity of the molecules is proportional to Vt, the diffusion 
coefficient increases with temperature as T 312 if the collision 
cross-section may be regarded as constant. 

The following comment should be made concerning the fore- 
going derivation. In calculating j we have argued as if only one 
gas were present, whereas in reality there is a mixture of two 
gases. It is therefore, strictly speaking, uncertain to which of the 
two gases the quantities o- and v pertain. Since only the order of 



332 DIFFUSION AND THERMAL CONDUCTION [XIV 

magnitude of the diffusion coefficient is being estimated, this 
point is unimportant if the molecules of the two gases are similar 
in mass and size, but it may become significant if there is a great 
difference between them. A more detailed discussion shows that 
in this case v must be taken as the greater of the thermal velocities 
(i.e. the velocity of the molecules of smaller mass), and cr as the 
greater of the cross-sections. 

As well as mutual diffusion of different gases, there can occur 
mutual diffusion of different isotopes of the same substance. 
Since the only difference between the isotopic molecules is the 
relatively slight difference in mass, this is a type of diffusion of 
gas molecules in their own gas, called self-diffusion. The difference 
in mass of the molecules here acts in practice only as a "label" 
whereby one molecule may be distinguished from others. 

The self-diffusion coefficient of a gas is given by the same 
formula 

D ~ vl, 

where there is now no problem as to the significance of the quan- 
tities which appear, since all refer to molecules of the only gas 
present. 

As examples, the following are the values of the diffusion 
coefficient in a number of gases at atmospheric pressure and 
0°C(incm 2 /sec): 

Hydrogen-oxygen mixture 0-70 

Carbon dioxide-air mixture 0-14 

Water vapour-air mixture 0-23 

Nitrogen (self-diffusion) 0-18 

Oxygen (self-diffusion) 0-18 

Carbon dioxide (self-diffusion) 0-10 

Diffusion in gases occurs much more rapidly than in liquids. 
For comparison we may mention that the diffusion coefficient of 
sugar in water (at room temperature) is only 0-3 x 10 -5 cm 2 /sec, 
and that of sodium chloride in water is 1 • 1 X 10 -5 cm 2 /sec. 

It is of interest to compare the true distance travelled by gas 
molecules in their thermal motion with their mean directed 
displacement in diffusion. For instance, air molecules under 



§113] DIFFUSION AND THERMAL CONDUCTION IN GASES 333 

normal conditions travel' distances of the order of 5 X 10 4 cm/sec. 
The diffusion displacement per second is, in order of magnitude, 
only V(Dt) ~ V(0-2 X 1) ~ 0-5 cm. 

The determination of the thermal conductivity of a gas does not 
require any essentially new calculations: we need only make use 
of the analogy noted in §109 between the processes of thermal 
conduction and diffusion, whereby thermal conduction appears 
as a "diffusion of energy", with the thermal diffusivity x acting 
as the diffusion coefficient. In a gas, the two processes occur by 
the same mechanism, namely direct transport by gas molecules. 
We can therefore say that, in order of magnitude, the thermal 
diffusivity x is equal to the self-diffusion coefficient of the gas, i.e. 

X ~ vl. 

The thermal conductivity k is obtained by multiplying x by 
the specific heat of the gas per unit volume. This volume contains 
nlN gram-molecules of the gas (where N is Avogadro's number), 
and the volume specific heat is therefore nC/N , where C is 
the molar specific heat; there is no need to distinguish between 
C p and C v , since they are the same in order of magnitude. Thus 

k ~ x«C/N ~ vlnC/No, 
and substituting / ~ Una we have finally 

k ~ vClaN . 

The molar specific heat of a gas is independent of its density. 
We therefore arrive at a remarkable (and at first sight paradoxical) 
result: the thermal conductivity of a gas depends only on its 
temperature, and not on its density or pressure. 

The specific heat of a gas depends only slightly on the tem- 
perature, and the same is true of the cross-section. We can there- 
fore suppose that the thermal conductivity of a gas, like the 
thermal velocity v, is proportional to VI. In reality, the thermal 
conductivity increases somewhat more rapidly with temperature, 
because the specific heat usually increases and the cross-section 
usually decreases. 



334 DIFFUSION AND THERMAL CONDUCTION [XIV 

As examples, the following are the values of the thermal con- 
ductivity of some gases at 0°C (in J/cm.sec.deg): 

Chlorine 0-72 X 10~ 4 Air 2-41 X 10 -4 

Carbon dioxide 1-45 X lO" 4 Hydrogen 16-8 X 10~ 4 

§114. Mobility 

Let consider a gas containing a number of charged particles 
(ions). If this gas is placed in an electric field, an ordered motion 
in the direction of the field is superposed on the random thermal 
motion of the ions which they execute in common with the gas 
molecules. If the ions were completely free particles, they would 
move with steadily increasing velocity under the action of the 
applied field. In reality, however, the ions move freely only in 
the intervals between collisions with the other particles in the gas. 
In collisions the particles are randomly scattered, and so the 
ions essentially lose the directed velocity which they acquire 
between collisions. Thus a motion results in which the ions, on 
average, slowly move or drift in the direction of the field at a 
certain velocity u proportional to the field strength. 

The order of magnitude of this velocity is easily estimated as 
follows. An ion of charge e and mass m in an electric field E is 
subject to a force F = eE, which gives the ion an acceleration 
w = Fjm. The ion moves with this acceleration during the mean 
free time r, and acquires a directed velocity of the order u ~ wt. 
Putting r ~ Uv, where v is the velocity of thermal motion of the 
ions, we have 

u ~ Fl/mv. 

The drift velocity u acquired by the ions under the action of the 
external field is usually written in the form 

u = KF; 

the coefficient of proportionality K between the velocity and the 
force F acting on the ions is called the ion mobility. 

As examples, the values of the mobility at 20° C and atmos- 
pheric pressure are 

for H 2 + ions in H 2 gas 8-6 x 10 12 cm/sec. dyn, 
for N 2 + ions in N 2 gas 1 -7 x 10 12 cm/sec.dyn. 



§114] MOBILITY 335 

This means, for example, that a field of 1 V/cm will cause N 2 + 
ions in nitrogen to drift with a velocity 1-7 X 10 12 X 4-8 X lO' 10 X 
1/300 = 3 cm/sec. 

From the above estimate of the velocity u it is seen that 
K ~ l/mv. Comparing this with the diffusion coefficient for the 
same particles (ions) in the gas, D ~ Iv, we see that D ~ mv 2 K, 
and since mv 2 ~ kT we have 

D ~ kTK. 

We shall show that this relation between the diffusion coefficient 
and the mobility of the particles is in fact an exact equality. 

According to Boltzmann's formula, in a state of thermal equilib- 
rium the ion concentration in a gas in a constant external electric 
field (which we take to be in the direction of the x axis) is propor- 
tional to 

e -U(x)lkT 

where U(x) = —Fx is the potential energy of an ion in the field; it 
varies through the gas, increasing in the direction of the field. 
When a concentration gradient is present, however, a diffusion 
flux j — —D dcjdx occurs. Let the concentration c be defined as 
the number of ions per unit volume of the gas, in the form 

c = constant X e FxlkT , 

since dcldx = (F/kT)c, we have 

j=-cDF/kT. 

In a steady (equilibrium) state, however, there can be no transfer 
of material in the gas. Thus the diffusion flux j in the opposite 
direction to the field must just compensate the drift flux of the 
ions in the direction of the field, which is evidently cu = cKF. 
Equating the two expressions, we obtain 

D = kTK. 

This relation between the mobility and the diffusion coefficient, 
called Einstein's relation, has been derived here for gases, but is 



336 DIFFUSION AND THERMAL CONDUCTION [XIV 

in fact general. It applies to any particles dissolved or suspended 
in a gas or a liquid and moving under the action of any external 
field (electric or gravitational). 

§115. Thermal diffusion 

In discussing diffusion in a gas mixture, we have so far tacitly 
assumed that the temperature (and pressure) of the gas is every- 
where the same, so that diffusion occurs only because of the con- 
centration gradient in the mixture. In reality it is found that a 
temperature gradient also may bring about diffusion. In a non- 
uniformly heated mixture, diffusion occurs even if the composi- 
tion is uniform; the difference in the thermal motion of the mole- 
cules of different components of the mixture (i.e. the difference in 
their thermal velocities and cross-sections) has the result that the 
two components appear in different proportions in the numbers of 
molecules crossing any area in the direction of the temperature 
gradient and in the opposite direction. The occurrence of a 
diffusion flux under the action of a temperature gradient is called 
thermal diffusion. This phenomenon is particularly important in 
gases, which we shall henceforward consider, but exists in 
principle in liquid mixtures also. 

The diffusion flux j T in thermal diffusion is proportional to the 
temperature gradient in the gas, and is customarily written in the 
form 

jT ~ Dr Tdx 

The quantity D T is called the thermal diffusion coefficient. Here 
we should specify exactly what is meant by the flux j' r (unlike the 
case of ordinary diffusion, where the coefficient D is independent 
of the way in which the flux is defined); we shall not pause to do 
this, however. Whereas the diffusion coefficient D is always posi- 
tive, the sign of the thermal diffusion coefficient is by its nature 
indeterminate, depending on which component of the mixture is 
considered. 

When the concentration of either component of a mixture 
tends to zero, the thermal diffusion coefficient must become zero, 
since there is of course no thermal diffusion in a pure gas. Thus 
the thermal diffusion coefficient depends considerably on the 



§115] THERMAL DIFFUSION 337 

concentration of the mixture, again unlike the ordinary diffusion 
coefficient. 

Because of thermal diffusion, concentration differences occur 
between regions at different temperatures even in a gas mixture 
of initially uniform composition. These concentration differences 
in turn cause ordinary diffusion, which acts in the opposite direc- 
tion, i.e. tends to annul the concentration gradient that has been 
formed. Under steady conditions, when a constant temperature 
gradient is maintained in the gas, these two opposite effects finally 
bring about a steady state in which the two fluxes compensate 
each other; in this state there is a certain difference in composi- 
tion between the "hot" and "cold" ends of the gas. 

Let us consider the simple case where the molecules of the two 
gases in the mixture are so different in mass that the thermal velo- 
city of the "heavy" molecules is small compared with that of the 
"light" molecules. The light molecules, on colliding with the 
heavy molecules, which may be regarded as at rest, rebound from 
them elastically, and under these conditions we need consider 
only the diffusion transport of the lighter component of the 
mixture. 

Let «! be the number of molecules of the light component per 
unit volume, and v t their thermal velocity. The flux of this com- 
ponent in the jc direction can be estimated by taking the difference 
between the values of the product n x v x at the points x — l x and 
x+l u where l x is the mean free path of the molecules. As in 
§113, this difference may be replaced by 

—lid{riiV^)ldx. 

Hence we see that the transfer of material ceases (i.e. a steady 
state is established) when the product n^Vi becomes constant 
throughout the gas. But «! = en, where c is the concentration 
of the light component, and n the total number of molecules per 
unit volume, which is equal to plkT. Since the total pressure 
p of the gas is everywhere the same, and the thermal velocity v t 
is proportional to Vl, the condition that n^ is constant implies 
that the ratio c/Vl is constant. In other words, in the steady 
state the concentration of the light component is greater in the 
hotter regions. 

This is in fact the way in which the composition varies in the 
majority of cases: the lighter gas usually accumulates at the "hot" 



338 



DIFFUSION AND THERMAL CONDUCTION 



[XIV 



end. This rule is not completely general, however, and the mass 
of the molecules is not the only factor which determines the 
direction of thermal diffusion. 

The phenomenon of thermal diffusion is utilised for the separa- 
tion of gas mixtures, and in particular for the separation of iso- 
topes. The principle of the method is clear from the construction 
of a simple "separating column" operating by thermal diffusion 
(Fig. 130). This consists of a long vertical glass tube with an 
electrically heated wire along its axis; the walls of the tube are 
cooled. The hot gas mixture rises along the axis and the cold 



i ! 



Fig. 130. 

mixture descends along the walls. At the same time a process 
of radial thermal diffusion occurs, as a result of which one com- 
ponent of the mixture (usually that of greater molecular weight) 
diffuses predominantly to the periphery, and the other to the 
axis. The components are entrained by the descending and 
rising currents, and accumulate at the bottom and top of the tube 
respectively. 



§ 1 1 6. Diffusion in solids 

Diffusion can also occur in solids, but is an extremely slow 
process. The phenomenon can be observed, for example, by 
fusing gold on the end of a rod of lead and keeping it at a high 
temperature, say 300°C; even in 24 hours, the gold penetrates 
about a centimetre into the lead. 



§116] DIFFUSION IN SOLIDS 339 

There is, of course, also self-diffusion in solids — the mutual 
diffusion of isotopes of the same substance. This can be observed 
by means of radioactive isotopes. If, for example, a quantity of 
a radioisotope of copper is placed on the end of a copper rod and 
the rod is later cut into pieces, the radioactivity of the pieces 
gives an idea of the diffusion of the isotope. 

The slowness of diffusion in solids is entirely understandable 
in view of the nature of the thermal motion of the atoms in them. 
In gases, and even in liquids, the random thermal motion of the 
molecules includes a "translational component", the molecules 
moving through the volume occupied by the body. In solids, 
however, the atoms are almost always near certain equilibrium 
positions (the lattice points) and execute small oscillations about 
these; such a motion can not lead to any general movement of 
the atoms through the body, nor therefore to diffusion. Only 
atoms which leave their positions in the lattice and move to other 
lattice points can take part in diffusion. 

However, each atom in a solid is surrounded by a potential 
barrier. An atom can leave its position only by surmounting this 
barrier, and to do so it must have sufficient energy. A similar 
situation has been discussed in connection with the rates of 
chemical reactions (§91), where we saw that the number of 
molecules able to react is proportional to an "activation factor" 
of the form 



The number of atoms which can participate in diffusion will also 
be proportional to such a factor, and therefore so will the diffusion 
coefficient. The value of the activation energy E per atom (E/N ) 
is usually between a fraction of an electron-volt and several 
electron-volts. For example, in the diffusion of carbon in iron 
E is about lOOkJ/mole (i.e. about 1 eV per atom); for the self- 
diffusion of copper, E is about 200 kJ/mole (about 2 eV per atom). 
Thus the diffusion coefficient in solids increases very rapidly 
with increasing temperature. For instance, the diffusion coeffi- 
cient of zinc in copper increases by a factor of 10 14 when the 
temperature is raised from room temperature to 300°C. One of 
the most rapidly diffusing pairs of metals is gold and lead, which 
have already been mentioned above. The diffusion coefficient of 



340 DIFFUSION AND THERMAL CONDUCTION [XIV 

gold in lead at room temperature is 4 x 10 -10 cm 2 /sec; at 300°C it 
is 1 x 10~ 5 cm 2 /sec. These figures also show the slowness of the 
diffusion process in solids. 

The acceleration of diffusion by increasing the temperature is 
the basis of the annealing of metals: in order to make the com- 
position of an alloy homogeneous, it is held for a considerable 
time at a high temperature. The same method is used to relax 
internal stresses in metals. 

In solid solutions of the interstitial type, the solute atoms 
occupy positions in the "gaps" between atoms at the original 
lattice points. The diffusion in such solutions (e.g. of carbon in 
iron) takes place simply by the movement of the interstitial 
atoms from one gap to another. In substitution-type solutions, 
however, in an ideal crystal, all the available places are occupied; 
diffusion in such an ideal crystal would have to take place by 
simultaneous exchange of positions of two different atoms. In 
an actual crystal, there are always unoccupied places or vacan- 
cies, as already mentioned in §105. These play an important part 
in the actual mechanism of diffusion, which occurs by atoms from 
adjoining occupied lattice points "jumping" to the vacant 
positions. 



CHAPTER XV 

VISCOSITY 



§117. The coefficient of viscosity 

Let us consider a flow of liquid (or gas) in which the velocity 
of flow is different at different points. This is not an equilibrium 
state, and processes will occur which tend to equalise the veloci- 
ties of flow. Such processes are called internal friction or vis- 
cosity. Just as there is a heat flux from the hotter to the colder 
parts of a medium in thermal conduction, so in internal friction 
the thermal motion of the molecules causes a transfer of momen- 
tum from the faster to the slower regions of the flow. 

Thus the three phenomena of diffusion, thermal conduction 
and viscosity have analogous mechanisms. In all three there is an 
equalisation of a property of the body (composition, temperature, 
or velocity of flow) if this property is originally not uniform 
through the body; this brings about an approach to a state of 
thermal equilibrium. In all three cases this is achieved by a 
molecular transport of some quantity from one part of the body 
to another. In diffusion there is a transport of number of particles 
of the various components of the mixture, in thermal conduction 
a transport of energy, and in internal friction a transport of 
momentum. All these effects are therefore often combined under 
the general name of transport phenomena. 

Let us suppose that a liquid is flowing in the same direction at 
all points, i.e. that the flow velocity vector u has the same 
direction throughout the flow, and suppose also that the mag- 
nitude u of the velocity varies in only one direction, perpendicular 
to that of the velocity, the direction of its variation being taken as 
that of the jc axis : u = u(x). 

By analogy with the diffusion flux and the heat flux, we can 
define the momentum flux II as being the total momentum trans- 
ported per unit time in the positive direction of the x axis across 
unit area perpendicular to that axis. In exactly the same way as 

341 



342 VISCOSITY [xv 

for the other transport processes, we can say that the momentum 
flux is proportional to the gradient of the flow velocity u: 

n = — 7] du/dx. 

The quantity 17 is called the coefficient of viscosity or simply the 
viscosity of the medium. 

The dimensions of the flux II are those of momentum divided 
by area and time, i.e. [II] = g/cm.sec 2 . The dimensions of du/dx 
are 1/sec. Hence 

[17] = g/sec.cm. 

The unit of viscosity in the CGS system is the poise (P). 

The viscosity determines the rate of transport of momentum 
from one point in the flow to another. The velocity is equal to the 
momentum divided by the mass. The rate of equalisation of the 
flow velocity is therefore determined by the quantity iq/p, where 
p is the density, i.e. the mass of the liquid per unit volume. The 
quantity v = 77/p is called the kinematic viscosity, whereas 17 
itself is called the dynamic viscosity. It is easily seen that 

[v] = cm 2 /sec, 

i.e. v has the same dimensions as the diffusion coefficient and the 
thermal diffusivity; the kinematic viscosity is a kind of diffusion 
coefficient for velocity. 

Let us suppose that a liquid flows in contact with a solid surface; 
for example, along the walls of a pipe. Between the surface of a 
solid and any actual liquid (or gas) there always exist forces of 
molecular cohesion which have the result that the layer of liquid 
immediately adjoining the surface is entirely brought to rest and 
"adheres" to the surface. Thus the flow velocity is zero at the 
wall, and increases away from the wall into the liquid; as a result 
of viscosity, there then occurs a flux of momentum from the liquid 
towards the wall. 

As we know from mechanics, the change in the momentum of a 
body per unit time is the force acting on the body. The momentum 
II transported through unit area per unit time and ultimately 
transferred from the liquid to the wall represents the frictional 



§118] VISCOSITY OF GASES AND LIQUIDS 343 

force exerted on unit area of the solid wall by the liquid flowing 
past it. 

The following comment should be added concerning the simple 
formula for II given above. Although the formal analogy already 
mentioned exists between the phenomena of diffusion, thermal 
conduction and viscosity, there is also an important difference 
between them, due to the fact that concentration and tempera- 
ture are scalar quantities, whereas velocity is a vector. Here we 
have taken only the simple case where the velocity is every- 
where in the same direction; the formula given above for II is 
valid only in this case. The impossibility of applying this for- 
mula when the direction of the velocity u is different at different 
points is evident from the example of a liquid rotating uniformly 
as a rigid body together with a cylindrical vessel, about the axis 
of the vessel. The circular velocity of the liquid particles increases 
with distance from the axis. There is, nevertheless, no flux of 
momentum, i:e. no frictional forces, in the liquid; a uniform rigid 
rotation of the liquid (if there is no friction in the suspension of 
the vessel) does not affect thermal equilibrium and could continue 
indefinitely without the velocity's becoming uniform. 

§118. Viscosity of gases and liquids 

The viscosity of a gas may be estimated from the fact that inter- 
nal friction, thermal conduction and self-diffusion all occur in a 
gas by the same molecular mechanism. In this case the quantity 
analogous to the diffusion coefficient is the kinematic viscosity 
v = 7)1 p. We can therefore say that, for a gas, all three quantities 
v, x and D are of the same order of magnitude, and thus we have 
v ~ vl. The gas density p = nm, where m is the mass of a mole- 
cule and n the number of molecules per unit volume; hence we 
have for the viscosity r) = vp the expression 

17 ~ mnvl ~ mvlcr, 

where a is the collision cross-section. 

We see that the viscosity, like the thermal conductivity, is 
independent of the pressure of the gas. Since the thermal velocity 
v is proportional to Vr, we may suppose that the viscosity of 
the gas is also proportional to the square root of the temperature. 
This conclusion is, however, valid only if we can regard the 



344 VISCOSITY [xv 

collision cross-section cr as constant. It has been mentioned in 
§112 that the cross-section increases somewhat with decreasing 
temperature. Accordingly, the viscosity decreases with decreas- 
ing temperature more rapidly than VT. 

The extent to which the approximate equality of the coeffi- 
cients v, x and D is maintained for gases can be seen, for example, 
from their values for air at 0°C: the kinematic viscosity v = 0-13, 
the thermal diffusivity x = 0'19, and the self-diffusion coefficient 
of nitrogen and oxygen D = 0- 18. 

The following are the values of the viscosities of some gases 
and liquids at 20°C: 





i7 


V 




(g/sec.cm) 


(cm 2 /sec) 


Hydrogen 


0-88 xlO- 4 


0-95 


Air 


1-8 X10~ 4 


0150 


Benzene 


0-65 


0-72 


Water 


0010 


0010 


Mercury 


0-0155 


0-0014 


Glycerine 


150 


12-0 



It is interesting to note that, whereas the dynamic viscosity of 
water is considerably greater than that of air, the reverse is true 
for the kinematic viscosity. 

The viscosity of a liquid usually decreases with increasing 
temperature; this is reasonable, since the relative motion of the 
molecules becomes easier. In liquids of low viscosity, such as 
water, the decrease is appreciable but not very great. There are, 
however, liquids, mainly organic (such as glycerine), whose 
viscosity decreases very rapidly with rising temperature. For 
example, a rise in temperature of 10° (from 20 to 30°C) causes 
the viscosity r) of water to decrease only by 20%, whereas that 
of glycerine decreases by a factor of 2-5. The decrease in vis- 
cosity of such liquids takes place exponentially, in proportion to 
a factor of the form e~ EIRT ; for glycerine, E « 65 000 J/mole. As 
we already know (cf. §116), this law of temperature dependence 
signifies that the occurrence of the process (in this case the 
relative motion of the molecules) requires the overcoming of a 
potential barrier. 

When the temperature decreases, a viscous liquid rapidly 
congeals into an amorphous solid. It has already been men- 



§119] poiseuille's formula 345 

tioned in §52 that the difference between a liquid and an amor- 
phous solid is purely quantitative. For example, rosin is a solid 
at room temperature, but even at 50-70°C it behaves as a fluid 
of high but measurable viscosity, 10 6 to 10 4 P; for comparison we 
may note that the consistency of honey or syrup corresponds to a 
viscosity of about 5 x 10 3 P. 

The mechanical properties of liquids such as glycerine and 
rosin are interesting in another respect also. The characteristic 
difference between a solid and a liquid is that the solid resists a 
change in shape (has a shear modulus) but the liquid does not. 
We may say that the molecular structure of a liquid is instantly 
"adjusted" to a change in shape; in typical liquids, this occurs in 
a time of the order of the periods of thermal vibration of the 
molecules (10~ 10 to 10 _12 sec). In liquid rosin, however, this 
"adjustment" requires a longer time, and when the deformation 
varies very rapidly it may be unable to occur; in rosin at 50-70°C 
the characteristic time is 10~ 4 to 10~ 5 sec. Thus such a substance 
behaves as an elastic solid with a certain shear modulus under a 
very rapidly changing external action (due to sound waves for 
example), but with respect to a slowly varying action it behaves 
as an ordinary liquid with a certain viscosity. 

§119. Poiseuille's formula 

We may use the formula II = — 17 duldx to solve a number of 
simple problems relating to the flow of a viscous liquid. 

Let us first calculate the frictional force between two parallel 
solid planes in relative motion, with a liquid of viscosity 77 in 
the space between them. Let u be the velocity of this motion, 




Fig. 131. 



and h the distance between the planes; in Fig. 131 the lower 
plane is at rest and the upper plane moves with velocity u . The 
liquid adjoining the walls is carried along by them, so that the 
velocity of the liquid is zero and u at the lower and upper walls 



346 VISCOSITY [XV 

respectively. In the region between the walls, the velocity u 
varies linearly: 

u = (ujh)x, 

where x is the distance from the lower wall; this result is derived 
in the same way as in the exactly similar problem of thermal 
conduction in a plane layer (§1 10). The required frictional force 
acting on unit area of each of the solid planes and tending to 
slow down their relative motion is given by the momentum flux II, 
as described in §1 17; this is 

II = u r)lh, 

and is thus proportional to the relative velocity u of the planes 
and inversely proportional to the distance between them. 

Let us next consider the flow of liquid in a cylindrical tube of 
radius a and length L, with different pressures p x and p 2 main- 
tained at the ends of the tube; the liquid then flows along the 
tube under the action of the pressure difference b.p = p 2 —p x . 
The flow velocity u of the liquid is everywhere along the axis 
of the tube, and its magnitude varies in the radial direction 
(perpendicular to the axis), depending on only one coordinate, the 
distance r from the axis. We can therefore write the momentum 
flux transported radially as 

II = — t] duldr. 

Let us consider a volume of liquid bounded by a cylindrical 
surface of radius r within the tube and coaxial with it. The total 
flux of momentum through this surface (whose area is lirrL) is 

InrLU = — lirrLt] duldr. 

This is the frictional force exerted on the volume of liquid in 
question by the remaining liquid, and is balanced by the force 
due to the pressure difference between the ends of the cylinder, 
which is TriPkp. Equating these forces, we obtain 

duldr = -{rl2Lri)kp, 

whence 

u = — {r^lALri) Ap + constant. 



§119] poiseuille's formula 347 

The arbitrary constant is determined from the condition that 
the velocity is zero at the surface of the tube, i.e. for r = a. Thus 
we have finally 

M = (A/?/4LTj)(a 2 -r 2 ). 

Thus a liquid flowing in a tube has what is called a parabolic 
velocity profile: the velocity varies quadratically from zero at 
the wall to a maximum value (w max = a 2 Ap/4Lr}) on the axis of 
the tube (Fig. 132). 



3 



Fig. 132. 

Let us determine the mass M of liquid leaving the tube per 
unit time. If V(r) denotes the volume of liquid leaving per unit 
time through the cylinder of radius r, the differential of this 
function is evidently 

dV(r) = u(r)dS, 

where u{r) is the velocity of the liquid at a distance r from the 
axis, and dS is the area of an annulus of radius r and width dr. 
Since dS = lirrdr, we have 

dV{f) = lirrudr 

= (7rAp/2Lr 1 )(a 2 -r 2 )rdr 
= (irApl4Lr))(a 2 -r 2 )d(r 2 ). 

V{r) = (TrApMLrjXaV-ir 4 ); 



Hence 



the arbitrary constant is taken as zero, since we must have 
V(0) = 0. The total volume of liquid leaving the tube per unit 



348 VISCOSITY [xv 

time is equal to the value of V(r) for r = a. Multiplying this by 
the density p of the liquid, we find the required mass: 

M = (Trkpl8Lp)a 4 . 

This is called Poiseuille's formula. We see that the quantity of 
liquid leaving the tube is proportional to the fourth power of the 
tube radius. 

The examples discussed above relate to steady flow of a liquid, 
in which the velocity of the liquid at any point in the flow is 
constant in time. One example of non-steady motion may be 
mentioned here. Let us assume that a disc immersed in a liquid 
executes torsional oscillations in its plane; the liquid entrained 
by the disc also oscillates. These oscillations are, however, 
damped with increasing distance from the disc, and the question 
arises of the order of magnitude of the distance at which an 
appreciable damping occurs. This question is formally equivalent 
to the one discussed in §111 concerning thermal oscillations 
caused by a plate with variable temperature. The required 
"penetration depth" L of the oscillatory motion in the liquid is 
obtained by replacing the thermal diffusivity x> in the formula 
derived in § 1 1 1 , by the kinematic viscosity v of the liquid: 

L ~ V(iVa)), 
where <u is the frequency of the oscillations. 

§ 1 20. The similarity method 

Some simple problems of the motion of a liquid have been 
discussed above. In more complex cases, an exact solution of 
the problem usually involves very great mathematical difficulties, 
and is as a rule impossible. For example, it is not possible to 
give a general solution of the motion through a liquid of a body 
having even such an apparently simple form as that of a sphere. 

Consequently, in various problems of the motion of a liquid, 
great importance attaches to simple methods based on con- 
sideration of the dimensions of the physical quantities on which 
the motion can depend. 

Let us consider, for example, the uniform motion of a solid 
sphere through a liquid, and let the problem be to determine the 



§120] THE SIMILARITY METHOD 349 

drag force F experienced by the sphere. [Instead of regarding 
the body as moving through a liquid, we could take the precisely 
equivalent problem of flow of a liquid past a body at rest; this 
statement of the problem corresponds to observations of gas 
flow past bodies in a wind tunnel.] 

The physical properties of a liquid which determine its flow 
or the motion of bodies in it are described by only two quantities: 
the density p and the viscosity 17. In addition, in the case con- 
sidered the motion depends on the velocity u of the sphere and 
its radius a. 

Thus we have altogether four parameters, whose dimensions 
are as follows: 

[p] = g/cm 3 , [t?] = g/cm.sec, [u] = cm/sec, [a] = cm. 

From these we can form a dimensionless quantity as follows. 
The dimension g, first of all, can be eliminated in only one way: 
by dividing 77 by p to give the ratio v = Tj/p, with dimensions 
[v] = cm 2 /sec. Next, to eliminate the dimension sec, we divide 
u by v. \u\v\ = 1/cm. A dimensionless quantity is then obtained 
by multiplying the ratio u/v by the radius a. This quantity is 
denoted by the symbol Re: 

Re — uajv = pua/r), 

and is called the Reynolds number, it is a very important property 
of the motion of a liquid. Clearly any other dimensionless quantity 
can only be a function of the Reynolds number. 

Let us return now to the determination of the drag force. Its 
dimensions are g.cm/sec 2 . A quantity having these dimensions 
which can be formed from the same parameters is, for example, 
pu 2 a 2 . Any other quantity having the same dimensions can be 
written as a product of pu 2 a 2 and some function of the dimen- 
sionless Reynolds number. We can therefore say that the 
required drag force is given by a formula of the type 

F = pu 2 a 2 f(Ke). 

The unknown function /(Re) cannot, of course, be determined 
from dimensional considerations alone, but we see that by 
means of these considerations we have been able to reduce the 



350 VISCOSITY [XV 

problem of determining a function of four parameters (the 
force F as a function of p, 17, u and a) to a problem of determining 
a single function /(Re). This function may be found experi- 
mentally, for example. By measuring the drag on any one 
sphere in any one liquid and plotting from the results a graph of 
the function ./(Re), we can find the drag in the motion of any 
sphere in any liquid. 

The above arguments are general and apply, of course, to 
steady motion in a liquid of bodies of any form (not only spheri- 
cal). The quantity a in the Reynolds number must then be taken 
as some linear dimension for a body of given shape, and we are 
thus able to compare the flow round geometrically similar bodies 
which differ only in size. 

Motions which have the same value of the Reynolds number 
for different values of the parameters p, 17, u, a are said to be 
similar. The entire pattern of the motion of the liquid in such 
cases differs only in the scale of distances, velocities etc. 

Although for brevity we have spoken of liquids, the whole of 
the above discussion applies to gases. The only condition as- 
sumed to be satisfied is that the density of the medium (liquid or 
gas) does not undergo any appreciable change during the motion, 
and may therefore be regarded as constant; in such circum- 
stances the moving medium is said to be incompressible. Although 
from the ordinary point of view a gas is an easily compressible 
medium, the changes in pressure which occur in a gas during its 
motion are usually insufficient to cause any appreciable change 
in its density. The gas ceases to behave as an incompress- 
ible medium only at velocities comparable with that of sound. 

§121. Stokes' formula 

Let us again consider the drag F encountered by a body moving 
in a liquid (or gas). When the velocity is sufficiently small, the 
drag force is always proportional to the velocity. In order to 
derive such a relation from the formula 

F = p« 2 « 2 /(Re), 

we must suppose that at low velocities the function /(Re) is 
of the form constant/Re. This gives 

F = constant X rjau. 



§121] stokes' formula 351 

We see that, if the drag is proportional to the velocity, it neces- 
sarily follows that the drag is also proportional to the linear 
dimension of the body (and to the viscosity of the liquid). 

The determination of the proportionality coefficient in this 
relation requires more detailed calculations. When a sphere 
moves in a liquid, the constant is found to be 6tt, i.e. 

F = 67riqau, 

where a is the radius of the sphere. This is Stokes' formula. 

The above discussion enables us to state more precisely what 
is meant by a "sufficiently small" velocity for Stokes' formula to 
be valid. Since the form of the function /(Re) is in question, the 
required condition must relate to the values of the Reynolds 
number, and since the number Re is proportional to the velocity 
u (for a given size of the body), it is clear that the condition for 
the velocity to be small must be that the dimensionless number 
Re is small: 

Re = au\v < 1. 

Hence we see that the condition for the velocity to be "sufficiently 
small" is a relative one. The actual range of permissible velocities 
depends on the size of the moving body (and on the viscosity 
of the liquid). For very small bodies (e.g. fine particles, suspended 
in a liquid, in Brownian motion) Stokes' formula is valid even 
for velocities which in other respects could not be regarded as 
small. 

If a sphere moves in a liquid under the action of an external 
force P (for instance, the force of gravity, with allowance for the 
partial loss of weight in the liquid), a uniform motion will finally 
be established with a velocity such that the force P is just 
balanced by the drag force. If P = F, the velocity is given by 

u = Pldnat]. 

This formula is frequently used to determine the viscosity of 
a liquid from a measurement of the rate of fall of a solid sphere 
in it The viscosity may also be determined by means of Poiseuille's 



352 VISCOSITY [XV 

formula, by measuring the rate of outflow of a liquid from a pipe 
along which it is impelled by a given pressure difference. 

Stokes' formula is also the basis of a method of measuring the 
unit charge, first used by Millikan to measure the charge on the 
electron. In these experiments fine droplets produced by an oil 
spray were placed in the space between horizontal plates forming 
a plane capacitor. The droplets have a charge owing to electrifica- 
tion in the spraying process or absorption of ions from the air. 
By observing under a microscope the rate of fall of a droplet by 
the effect of its weight alone, we can use Stokes' formula to 
calculate the radius and hence the mass of the drop (whose 
density is known). Then, by applying a suitable potential differ- 
ence across the capacitor, we can bring the droplet to rest, the 
downward force of gravity being balanced by the upward electrical 
force on the charged droplet. Knowing the weight of the droplet 
and the electric field strength, we can calculate the charge on the 
droplet. Such measurements show that the charge on the droplets 
is always an integral multiple of a certain quantity, which is 
evidently the unit charge. 

§122. Turbulence 

The flow of a liquid in a pipe as described in § 1 1 9 is orderly and 
smooth: each liquid particle moves in a fixed straight line and the 
whole pattern of the flow is like the motion of various layers of 
liquid with different relative velocities. This kind of regular steady 
flow of a liquid is called laminar flow. 

It is found, however, that a liquid flow of such a kind continues 
to occur only when the Reynolds number is sufficiently small. 
For flow in a pipe, this number may be defined by the formula 
Re = udlv, where d is the diameter of the pipe and u the mean 
velocity of the liquid. For example, if the flow velocity is increased 
in a pipe of given diameter, a point is reached at which the nature 
of the flow changes completely. It becomes extremely dis- 
ordered, and instead of smooth lines the liquid particles describe 
tangled, meandering and continually changing paths. Such motion 
is said to be turbulent. 

The difference between the two types of motion appears very 
clearly if we observe the flow in a glass tube and introduce a small 
quantity of a coloured liquid into the flow through a capillary. 
At low velocities, the coloured liquid is carried along by the 



§122] TURBULENCE 353 

main flow as a thin straight filament. At high velocities, however, 
this filament is disrupted and the coloured liquid mixes rapidly 
and almost uniformly with the entire flow. 

If we follow the variation of the liquid velocity with time at 
any given point in a turbulent flow, we find irregular random 
fluctuations of the velocity about some mean value. The mean 
values of the velocity describe the pattern of motion of the liquid 
in which the irregular turbulent fluctuations or eddies are smoothed 
out. This averaged velocity is usually what is meant in speaking 
simply of the velocity of a turbulent liquid flow. 

Turbulent mixing of a liquid is a much more efficient means of 
transferring momentum than the process of molecular transfer 
by internal friction in a laminar flow. For this reason the velocity 
profile over the cross-section of a pipe in turbulent flow is con- 
siderably different from that in laminar flow. In the latter, the 
velocity gradually increases from the wall to the axis of the pipe, 
but in turbulent flow the velocity is almost constant over a large 
part of the cross-section of the pipe, falling rapidly in a thin layer 
adjoining the wall to the value zero which it must have at the 
wall itself. 

The unimportant role of viscosity in comparison with turbulent 
mixing also has more general consequences: the viscosity has 
no direct effect on the properties of turbulent flow, and these 
properties are therefore determined by a smaller number of 
quantities than in laminar flow, since these do not include the 
viscosity of the liquid. The possibilities of constructing different 
combinations having the same dimensions from the remaining 
quantities are much more restricted, and the application of the 
similarity method may therefore immediately give more specific 
results. 

Let us find, for example, the relation between the mean velocity 
u of flow in a pipe and the pressure gradient which brings about this 
flow (i.e. the ratio Ap/L, where Ap is the pressure difference 
between the ends of the pipe and L the length of the pipe). The 
quantity Ap/L has the dimensions g.cm~ 2 .sec -2 . The only combina- 
tion having these dimensions which can be constructed from the 
available quantities (the velocity u, the diameter d of the pipe 
and the density p of the liquid) is pw 2 / d. We can therefore assert 
that 

Ap/L = constant X piPId, 



354 VISCOSITY [xv 

where the constant is a pure number. Thus in turbulent flow in a 
pipe the pressure gradient is proportional to the square of the 
mean velocity, and not to the velocity itself as in laminar flow. 
[This law, however, is only approximately valid, since no account 
has been taken of the effect of the boundary layer at the wall, 
in which the velocity decreases very rapidly and the viscosity 
plays an important part.] 

It has already been mentioned that flow in a pipe becomes 
turbulent for sufficiently large Reynolds numbers. Experiment 
shows that, for this to occur, the Reynolds number must exceed 
1700. For smaller values of the Reynolds number, laminar flow 
is completely stable. This means that, when the flow is perturbed 
by some external agency (vibration of the pipe, roughness of the 
pipe inlet, etc.), the resulting deviations from smooth flow are 
rapidly damped. When Re > 1700, on the other hand, perturba- 
tions of the flow lead to disruption of the laminar condition and 
the appearance of turbulence. By means of special precautions 
to reduce the perturbations which inevitably occur, it is possible 
to postpone the transition to turbulent flow until still higher 
values of Re are reached, and laminar flow in a pipe has been 
observed even for Re = 50 000. 

Turbulence is a general feature of flow at very high Reynolds 
numbers. It occurs not only in flow in a pipe but also in flow of a 
liquid (or gas) past various solid bodies (or, what amounts to the 
same thing, motion of these bodies through a liquid). Let us 
consider the pattern of such flow in more detail. 

Because of the law of similarity discussed in §120, it is im- 
material what is the precise reason for the large value of the 
Reynolds number, whether the size a of the body is large, the 
velocity u is large, or the viscosity rj is small. In this sense we 
can say that for very large Reynolds numbers the liquid behaves 
as if its viscosity were very low. This applies, however, only to a 
liquid flowing far from solid walls. Near the surface of a solid a 
thin boundary layer is formed in which the velocity decreases 
from the value corresponding to frictionless motion to the value 
zero corresponding to the adhesion of the viscous liquid to the 
wall. The thickness of the boundary layer decreases with in- 
creasing Reynolds number. Within this layer the velocity changes 
very rapidly, and hence the viscosity of the liquid is of decisive 
importance. 



§122] TURBULENCE 355 

The properties of the boundary layer lead to the important 
phenomenon of separation in flow past bodies. When a liquid 
flows past a body, it first of all moves along the front part of the 
body, which becomes wider in the direction of flow. The streams 
of liquid undergo a kind of compression, and their velocity 
accordingly increases and the pressure decreases, as follows 
from Bernoulli's equation (see §61). In flow along the rear part 
of the body, however, the streams expand as the body narrows, 
the velocity in them decreases, and the pressure correspondingly 
rises. Thus in this part of the flow the pressure increases in the 
direction of flow, i.e. the pressure difference opposing the motion 
of the liquid increases. This pressure difference arising in the 
main flow acts also on the liquid in the boundary layer and 
retards it. The liquid particles in the boundary layer move more 
slowly than in the main flow and begin to move even more slowly, 
until, when a point on the surface of the body is reached at which 
the pressure is sufficiently high, the particles in the boundary 
layer come to rest and begin to move in the opposite direction. 
Thus a reversed motion occurs near the surface of the body, 
despite the fact that the main flow continues to move in the 
same direction. At points still further along the surface of the 
body, the reverse flow becomes wider and ultimately displaces 
the main flow completely, which thus becomes separated from 
the surface. 

This motion with a reversed flow, however, is entirely unstable 
and immediately becomes turbulent. The turbulence extends 
forward along the flow and gives rise to a long strip of liquid in 
turbulent motion behind the body, called the turbulent wake; 
this is shown diagrammatically in Fig. 133. For example, this 




Fig. 133. 



occurs for a sphere when Re is greater than about 1000, Re being 
defined as du/v, where d is the diameter of the sphere. 



356 VISCOSITY [xv 

At very large Reynolds numbers the formation of the turbulent 
wake is the principal cause of the drag on the body moving in 
the liquid. Under these conditions we can again use dimensional 
arguments to determine the law of drag. The drag force F on 
a body (of given shape) can depend only on the size a of the body, 
its velocity u and the density p of the liquid, but not on its 
viscosity. From these three quantities only one combination 
having the dimensions of force can be derived, namely the 
product pu 2 a 2 . We can therefore say that 

F = constant X pu 2 a 2 , 

where the constant depends on the shape of the body. Thus, at 
very high Reynolds numbers the drag force is proportional to 
the square of the velocity (Newton's law of drag). It is also 
proportional to the square of the linear size of the body or, what 
is the same thing, to its cross-sectional area (which is itself 
proportional to a 2 ). Finally, the drag force is proportional to 
the density of the liquid. In the opposite case of small Reynolds 
numbers, it will be remembered that the drag is proportional 
to the viscosity of the liquid and independent of the density. 
Whereas at small Reynolds numbers the drag is determined by 
the viscosity of the liquid, when Re is large the effect of the 
inertia (mass) of the liquid becomes predominant. 

The drag at large Reynolds numbers depends very greatly on 
the shape of the body, which determines the point of separation 
of the flow and therefore the width of the turbulent wake. The drag 
due to the turbulent wake is smaller when the wake is narrower. 
This determines the choice of shape of a body such that the drag 
force is the least possible; such a shape is said to be streamlined. 

A streamlined body must be rounded at the front and drawn 
out at the back to run smoothly to a pointed end, as shown in 
Fig. 134 (this diagram may be regarded as showing the longi- 
tudinal cross-section of an elongated solid of revolution or as 
the cross-section of a "wing" of large span). The liquid flow along 




Fig. 134. 



§123] RAREFIED GASES 357 

such a body closes up smoothly at the pointed end, with no sharp 
turns anywhere; this eliminates the rapid rise of pressure in the 
direction of the flow. The flow is separated only at the actual 
point, and the turbulent wake is therefore extremely narrow. 

In connection with the drag at high velocities, it should be 
mentioned that the foregoing discussion refers only to velocities 
small compared with that of sound, so that the liquid may be 
regarded as incompressible. 

§123. Rarefied gases 

The. conclusions drawn in §§1 13 and 118 concerning transport 
processes in gases are valid only so long as the gas is not too 
rarefied: the mean free path of the molecules must be small 
compared with the size of the bodies considered (the vessel 
containing the gas, the bodies moving through the gas, etc.). 
However, even at a pressure of 10 -3 to 10 -4 mm Hg the mean 
free path is 10-100 cm, and is comparable with or even exceeds 
the usual dimensions of apparatus. A similar situation occurs in 
problems of space flight near the earth: even at an altitude of 
about 100 km the mean free paths of the particles in the ionised 
gas are several tens of metres. 

Here we shall apply the term rarefied to gases in which the 
mean free path of the molecules is large compared with the size 
of the bodies. This criterion depends not only on the state of the 
gas itself but also on the size of the bodies that are in question. 
A given gas may therefore behave as a rarefied gas in some 
conditions but not in others. 

Let us consider the transfer of heat between two solid plates 
heated to different temperatures and immersed in a gas. The 
mechanism of this process is totally different in rarefied and non- 
rarefied gases. In non-rarefied gases, the transfer of heat from 
the hotter to the colder wall occurs by the gradual "diffusion of 
energy" transferred from one molecule to another in collisions. 
But if the mean free path / of the gas molecules is large in com- 
parison with the distance h between the walls, the molecules in 
the space between the plates will scarcely ever collide with one 
another, and will move freely after reflection from one plate until 
they collide with the other. When scattered from the hotter plate, 
the molecules gain some energy from it, and they give up part of 
their energy on colliding with the colder plate. 



358 VISCOSITY [xv 

Under these conditions there is obviously no meaning in speak- 
ing of the temperature gradient in the gas between the plates, 
but by analogy with the expression q = — K dT/dx for the heat 
flux we can now define the "thermal conductivity" of a rarefied 
gas by the relation 

q = -k(T 2 - TJ/h, 

where T 2 — T t is the temperature difference between the plates. 
We can estimate this conductivity in order of magnitude directly 
by analogy with the expression derived in §113 for the ordinary 
thermal conductivity: 

k ~ vlnC/N . 

There is no need to repeat the derivation; we need only note that, 
since we now have collisions with the plates instead of between 
molecules, the mean free path in this formula must be replaced 
by the distance h between the plates: 

k ~ vhnC/N 

(C is the molar specific heat of the gas, v the thermal velocity of 
the molecules, and n the number of molecules per unit volume). 
Substituting n = p/kT and replacing the product N k by the gas 
constant R, we have 

k ~ phvClRT. 

We see that the "thermal conductivity" of a rarefied gas is 
proportional to its pressure, unlike the thermal conductivity of 
a non-rarefied gas, which is independent of the pressure. It must 
be emphasised, however, that this conductivity is no longer a 
quantity pertaining to the gas alone: it depends also on the 
distance h between the two bodies. 

The decrease in the thermal conductivity of a rarefied gas with 
decreasing pressure is the basis of the use of an evacuated space 
for thermal insulation, as for instance in Dewar vessels for keep- 
ing liquefied gases; these have double walls with the air evacuated 
from the space between. As the evacuation proceeds the ther- 



§123] RAREFIED GASES 359 

mal conductivity of the air at first remains constant, and begins 
to decrease rapidly only when the mean free path becomes 
comparable with the distance between the walls. 

Internal friction in rarefied gases behaves similarly. Let us 
consider, for example, two solid surfaces with a layer of rarefied 
gas between them and moving with relative velocity u. The 
"viscosity" of the gas is defined by the relation 

II = t)ulh, 

where II is the frictional force per unit area acting on the solid 
surfaces, and h the distance between them. Replacing the mean 
free path / by h in the formula -q ~ nmvl derived in §118, we 
obtain 

7} ~ nmvh, 

and putting n=pjkT and kT ~ mv 2 we have finally 

77 ~ ph/v. 

Thus the "viscosity" of a rarefied gas is also proportional to the 
pressure. Like the thermal conductivity, the viscosity depends 
not only on the properties of the gas itself but also on the charac- 
teristic dimensions which appear in a given problem. 

Let us use the above expression for 17 to estimate the drag F 
on a body moving in a rarefied gas. Here h must be taken as the 
linear size a of the body. The frictional force per unit area of 
the surface of the body is 

II ~ t]u\a ~ pu/v, 

where u is the velocity of the body. Multiplying this by the area 
S of the surface of the body, we obtain 

F ~ upSlv. 

Thus the drag exerted by a rarefied gas is proportional to the 
surface area of the body, unlike the drag in a non-rarefied gas, 
which is proportional to the linear size of the body. 



360 VISCOSITY [XV 

We may also mention some interesting effects relating to the 
outflow of a rarefied gas through small holes, of size much less 
than the mean free path of the molecules. This outflow, called 
effusion, is quite different from the ordinary outflow through 
large holes, where the gas flows out in a jet like a continuous 
medium. In effusion the molecules leave the vessel independently 
and form a "molecular beam" in which each molecule moves with 
the velocity with which it approached the hole. 

The rate of outflow of a gas in effusion, i.e. the number of 
molecules leaving the hole per unit time, is in order of magnitude 
Snv, where S is the area of the aperture. Since n = p/kT, v ~ 
V(A:77m), we have 

Snv ~ SplV(mkT). 

The rate of effusion thus decreases with increasing mass of the 
molecule. Thus, in effusion of a mixture of two gases, the out- 
flowing gas will be enriched in the lighter component. This is 
the basis of one common method of isotope separation. 

Let us now imagine two vessels containing gases at different 
temperatures 7\ and T 2 and connected by a small hole (or a 
tube of small diameter). If the gases were not rarefied, the 
pressures in the two vessels would become equal and the force 
exerted by each gas on the other at the hole would be the same. 
For rarefied gases, however, this is no longer true, since the 
molecules pass freely through the hole without colliding with 
one another. The pressures /? x and p 2 then take values such that 
the numbers of molecules passing through the hole in each 
direction are the same. According to the expression derived 
above for the rate of outflow, this means that the condition 

pJ\/T 1 = p 2 ly/T 2 

must hold. Thus different pressures are set up in the two vessels, 
the pressure being higher in the vessel with the higher temper- 
ature. This is called the Knudsen effect. In particular, it must 
be taken into account in the measurement of very low pressures; 
a difference in the temperatures of the gas under examination 
and the gas in the measuring instrument causes a corresponding 
difference in pressure. 



§124] SUPERFLUIDITY 361 

§124. Superfluidity 

It has already been mentioned that liquid helium is exceptional 
in its physical properties, being a "quantum liquid", whose 
properties cannot be understood on the basis of the concepts 
of classical mechanics. This is shown by the fact that helium 
remains liquid at all temperatures down to absolute zero (§72). 

Helium becomes liquid at 4-2°K. At about 2-2°K it remains 
liquid but undergoes a further transformation, a phase transition 
of the second kind (see §74). At temperatures above the trans- 
formation point liquid helium is generally called helium I, and 
below it helium II. 

Helium II has the following properties. Firstly, heat transfer 
is extremely rapid. A temperature difference between the ends 
of a capillary filled with helium II is very quickly eliminated, 
and helium II is in fact the best heat conductor known. This 
property, incidentally, explains the striking change that is seen 
on observing visually the transformation of helium I into helium 
II: the surface of the continuously boiling liquid suddenly 
becomes completely calm and smooth when the transition point 
is reached. The reason is that, because of the very rapid removal 
of heat from the vessel walls, the vapour bubbles typical of 
boiling are no longer formed there, and helium II evaporates 
only from its free surface. 

The fundamental and primary property of liquid helium, 
however, is that of superfluidity, discovered by P. L. Kapitsa. 
This refers to the viscosity of liquid helium. 

The viscosity of a liquid can be measured from its rate of flow 
through narrow capillaries, but in the present case this method 
is unsuitable, and a method is required which allows the flow 
of a greater quantity of liquid than can pass through a narrow 
capillary. This is achieved by an experiment in which helium II 
passes along a very narrow gap (about 0.5 /x wide) between two 
discs of ground glass (Fig. 135). Yet even under these conditions 
liquid helium exhibits no viscosity whatever, which shows that 
the viscosity is exactly zero. The absence of viscosity in helium 
II is what is referred to by the term "superfluidity". 

The superfluidity of helium II is directly responsible for the 
"creeping film" which it forms. When helium II is placed in 
two vessels separated by a partition, it spontaneously takes 
the same level in both vessels after a certain time. This transfer 



362 



VISCOSITY 



[XV 







-- v/;//;;/ ///;///////77r - 



Fig. 135. 

takes place along a thin film (a few hundred angstroms thick) 
which is formed by the helium on the walls and which acts as 
a siphon (Fig. 136). The mere fact that a film is formed is not 
a specific property of helium II. A film is formed by any liquid 
which wets the solid surface. In ordinary liquids, however, the 
formation of the film and its spreading over the surface occur 
extremely slowly because of the viscosity of the liquid. The 



O r\ 



\ 


♦ ♦ 


♦ 


\ 

1 


1 ♦ 




T 
\ 


^£K 


T 

♦ 


\ 





i 







T 














j 





J 


r _— _d 


h= 


-^^-^^^ - -^ >s --^— r -— "— - '— ^ 



Fig. 136. 

formation and movement of the film occur rapidly for helium II, 
however, because of its superfluidity. The velocity of the creeping 
film is several tens of centimetres per second. 

We have discussed the viscosity of helium as measured by 
the rate of flow of the liquid through a narrow gap. But the 
viscosity of a liquid can also be measured in another way. If a 
disc (or cylinder) suspended in the liquid executes torsional 
oscillations about its axis, the friction on the disc which retards 
its oscillations is a measure of the viscosity. It is found that in 



§124] SUPERFLUIDITY 363 

such measurements helium II shows a small but non-zero 
viscosity (of the order of 10 -5 poise). 

The theory (due to L. D. Landau) which gives an explanation 
of these paradoxical properties of liquid helium cannot be 
explained here before the principles of quantum mechanics 
have been stated. We shall, however, describe the remarkable 
physical picture which results from this theory. 

It usually appears self-evident that, in order to describe the 
motion of a liquid, it is sufficient to specify its velocity at every 
point. Yet this seemingly obvious statement is invalid for the 
motion of the quantum liquid helium II. 

Helium II is found to be capable of executing two motions 
simultaneously, so that, in order to describe the flow, the values 
of not one but two velocities at every point must be specified. 
The situation may be visualised by regarding helium II as a 
mixture of two liquid components which can move independently 
"through" each other without mutual friction. In reality, however, 
there is only one liquid, and it must be emphasised that the 
"two-fluid" model of helium II is only a convenient way of 
describing the phenomena which occur in it. Like any description 
of quantum phenomena in classical terms, it is not completely 
adequate— as is reasonable, since our intuitive ideas reflect what 
we encounter in ordinary life, whereas quantum phenomena 
usually appear only in the microscopic world inaccessible to 
our direct perception. 

Each of the two motions simultaneously occurring in liquid 
helium involves the displacement of a certain mass of liquid. In 
this sense we can speak of the densities of the two "components" 
of helium II, although it must again be emphasised that this 
terminology does not at all signify that the atoms of the sub- 
stance can actually be divided into two classes. Each of the 
two motions is a collective motion of a large number of the same 
liquid atoms. 

The two motions have entirely different properties. One 
motion takes place as if the corresponding "component" of the 
liquid had no viscosity; this is called the superfluid component. 
The other {normal) component moves like an ordinary viscous 
liquid. 

Another important difference between the two types of motion 
in helium II is that the normal component transports heat in 



364 VISCOSITY [XV 

its motion, but the superfluid motion is not accompanied by 
any transport of heat. In a certain sense we may say that the 
normal component is the heat itself, which in liquid helium 
becomes independent and separate from the mass of the liquid, 
and becomes able to move relative to a "background" that is at 
absolute zero. This picture is radically different from the usual 
classical idea of heat as a random motion of atoms, inseparable 
from the whole mass of the substance. 

These concepts give an immediate explanation of the principal 
results of the experiments described above. First of all, they 
eliminate the contradiction between the measurements of the 
viscosity of the liquid from the friction experienced by a rotating 
disc and from the flow of liquid through a gap. In the first case 
the disc is retarded because, when it rotates in liquid helium, it 
undergoes friction with the "normal" part of the liquid, and the 
viscosity of this component is essentially what is measured. In 
the second case, the superfluid part of the helium flows through 
the gap, while the normal component, which has a viscosity, is 
retarded and "percolates" very slowly through the gap. Thus, 
in this experiment, the zero viscosity of the superfluid component 
is observed. 

But since the superfluid motion does not transfer heat, the 
outflow of helium through the gap causes a kind of filtering off 
of liquid without heat, the heat remaining in the vessel. In the 
ideal limiting case of a sufficiently narrow gap, the liquid leaving 
would be at absolute zero. In an actual experiment its temper- 
ature is not zero, but is lower than that of the vessel. For example, 
by driving helium II through a porous filter its temperature can 
be lowered by 0-3 to 0-4°, and this is a large amount when the 
temperature is only 1-2°K. 

The very high rate of heat transfer in helium II also finds a 
natural explanation. Instead of the slow process of molecular 
transport of energy in ordinary thermal conduction, we here 
have a rapid process of heat transfer by the normal component 
of the liquid. The relation between the process of heat transfer 
in helium II and the occurrence of motion in it is clearly shown 
by the following experiment. A light vane is placed in front of 
a hole in a small vessel filled with liquid helium (and immersed 
in liquid helium) (Fig. 137). When the helium in the vessel is 
heated, the vane is deflected. This occurs because heat flows 



§124] 



SUPERFLUIDITY 



365 



from the vessel as a stream of the viscous normal component, 
which deflects the vane in front of the hole. In the opposite 
direction there is an inflow of the super-fluid component, so that 
the actual quantity of liquid in the vessel is unchanged and the 
latter remains full. The superfluid component has no viscosity 
and does not move the vane as it flows past it. 

The presence of two "components" in helium II is shown 
directly in an experiment based on the following idea. When a 
cylindrical vessel containing liquid helium rotates, only part of 
the liquid should be carried round, namely the "normal" part, 
which undergoes friction against the walls; the "superfluid" 




Fig. 137. 




part should remain at rest. In an actual experiment, the rotation 
of the vessel is replaced by torsional oscillations of a stack of 
numerous thin discs; this increases the area of the surface which 
entrains the liquid. 

At temperatures above the transition point the liquid (helium I) 
is entirely in the normal state and is entirely carried round by 
the rotating walls. At the transition point a qualitatively new 
property of the liquid first appears, namely the occurrence of 
the superfluid component, and this is the nature of the phase 
transition of the second kind in liquid helium. As the temperature 
decreases further, the fraction of the superfluid component 
becomes greater, and at absolute zero the liquid should become 
entirely superfluid. Figure 138 shows the form of the temperature 
dependence of the ratio of the density p n of the normal component 
of the liquid helium to the total density p of the liquid (the sum 
of the normal and superfluid densities p„ and p s is, of course, 
always equal to the total density p). 



366 VISCOSITY [XV 

Finally, let us consider a phenomenon in liquid helium related 
to the propagation of sound waves in it. Sound waves in an 
ordinary liquid consist of periodic compressions and rarefactions 
propagated through the liquid, in which each particle executes 
an oscillatory motion about the mean equilibrium position with 
a periodically varying velocity. In helium II, however, two 
different motions can take place simultaneously with different 
velocities. There are therefore two quite distinct possibilities 
for the motion in a sound wave. If the two components of the 
liquid execute an oscillatory motion in the same direction, 
moving as it were in unison, we have a sound wave of the same 
kind as in an ordinary liquid. 

The two components may also, however, execute oscillations 
in opposite directions, moving "through" each other, so that 
the masses transported in each direction are almost exactly 
equal. In such a wave (called a second sound wave) there are 
almost no compressions and rarefactions of the liquid as a whole. 
Instead, there are periodic oscillations of temperature in the 
liquid, since the mutual oscillations of the normal and superfluid 
components are essentially oscillations of heat relative to the 
"superfluid background". Thus the second sound wave is a kind 
of "thermal wave", and it is therefore clear that the source 
needed to create such a wave is a heater whose temperature 
varies periodically. 

The whole of the foregoing discussion has referred simply to 
liquid helium. It is necessary to make explicit that this discussion 
relates only to one of the isotopes of helium, namely the common 
isotope He 4 . There exists also another much rarer isotope, He 3 . 
By the methods of nuclear physics it is possible to separate this 
isotope in quantities sufficient for liquefaction and experiments 
with the liquid. It is again a "quantum liquid", but its properties 
are entirely different, and in particular it is not a superfluid. 
Although the two isotopes of helium are chemically identical, 
there is a very important difference between them, due to the 
fact that the nucleus of the He 4 atom comprises an even number 
of particles (protons and neutrons), and that of He 3 an odd 
number. This difference has the result that the quantum properties 
of the two substances are entirely different, and so in turn brings 
about the difference in the physical properties of the liquids 
which they form. 



INDEX 



Absolute temperature 145 
Absolute zero 147 
Acceleration 10 

angular 74 

in circle 11-12 

dimensions of 17 

due to gravity 60 

normal 12 

tangential 12 

units of 17 
Action and reaction, law of 14 
Activation energy 268 
Activation factor 268 
Adiabatic processes 178-82 
Adsorption 279-82 
Aggregate states of matter 152,1 97 
Allotropy 218 
Amplitude of oscillation 87 
Angular acceleration 74 
Angular frequency 87 
Angular momentum 36 

conservation of 37,40 

rotational 72-73 
Angular velocity of rigid body 66-68 
Anisotropy 132 
Atomic number 1 06 
Atomic weight 105,110 
Atoms 105-11 
Avogadro's law 155 
Avogadro's number 105 
Axis of symmetry 115 

rotary-reflection 116 

screw 130 
Azeotropic mixture 243 



Barometric formula 159 
Beats 102 

Bending of rods 297-8 
Bernoulli's equation 186 
Boiling 206-7,241-6 



Boltzmann factor 1 63 
Boltzmann's constant 146 
Boltzmann' s formula 158 
Boyle's law 156 
Bravais lattices 121-9 
Brownian motion 144 
Buffer solution 266 
Bulk modulus 300 



Capillarity effects 276, 285-9 
Carnot cycle 191 
Catalysis 270 

Celsius scale of temperature 1 47 
Central field, motion in 40-43 
Centre of gravity 7 8 
Centre of inversion 115 
Centre of mass 9, 78 

frame of reference 1 
Centre of symmetry 115 
Centrifugal energy 83 
Centrifugal force 83 
CGS system of units 1 8 
CGSE system of units 45, 50 
Chain reactions 272-5 
Charge, see Electric charge 
Charles' law 157 
Chemical elements 105 

lattices of 134-7 
Chemical equilibrium 254-6 
Chemical reactions 252-75 
Classical mechanics 5 
Clausius-Clapeyron equation 203 
Close packing 135 
Closed system 5 
Collisions 3 1 

elastic 31-36 

inelastic 3 1 
Colloidal solutions 291-3 
Compressibility 174, 300 
Compression of solids 294, 298-30 1 



367 



368 



INDEX 



Conduction, thermal 3 1 9-34 
Conductivity, thermal 320, 333-4, 

358 
Conservation 

of angular momentum 37,40 

of charge 47 

of energy 24 

of mass 8 

of momentum 6 
Conserved quantities 6 
Contact, angle of 282-5 
Convection 319 
Coriolis force 83 
Corresponding states, law of 215 
Cosmic velocity 

first 62 

second 64 
Coulomb's law 44 
Couple 78 

arm of 78 

moment of 78 
Critical point 207-9 

of mixing 234 
Critical pressure 207 , 209 
Critical temperature 207, 209 

of mixing 234 
Cross-section 329 
Crystal 

classes 132 

lattices 120-43 

modifications 218-21 

planes 139-42 

space groups 129-31 

systems 123-9 
Crystals 

biaxial 129 

defects in 308-12 

liquid 227-9 

mixed 234-6 

ordering of 225-7 

single 153 

structure of 120-43 

uniaxial 1 29 
Cyclic process 167 



Dalton'slaw 157 
Damping coefficient 95 
Defects in crystals 308-1 2 
Degrees of freedom 3,68 
Derived units 15 



Diffusion 318 

coefficient 319 
thermal 336 

flux 318 

in gases 330-3 

in solids 338-40 

thermal 336-8 
Diffusivity, thermal 320 
Dimensions 17-18,25 
Dislocations 309-12 
Dissociation 

constant 264 

degree of 264 

electrolytic 262 
Drag force 348-51,359 
Dulong and Petit's law 1 76 



Efficiency of heat engine 191-2 

Effusion 360 

Elastic 

deformations 306 

energy 297 

limit 306 

properties 296 

stresses 294 
Electric 

charge 44 
conservation of 47 
dimensions of 45 
unit 106 
units of 45 

field 46-56 
potential of 49 

flux 51 

lines of force 48 

tubes of force 52 
Electrical interaction 45-56 
Electrolytes 

strong 262-3 

weak 264-6 
Electrons 106-8 
Electrostatic 

potential 49-50 

units 45 
Enantiomorphism 131 
Endothermic reaction 252 
Energy 

centrifugal 83 

conservation of 24 

dimensions of 25 



INDEX 



369 



Energy (cont.) 

elastic 297 

internal 26-27 

of particle 24 

of rigid body 68-72 

total 24 

units of 25-26 

see also Kinetic energy; Potential 
energy 
Enthalpy 170 
Entropy 194-6 
Equalisation time 326-7 
Equation of motion 

of particle 1 3 

of rotating rigid body 74 
Equation of state 150 

for ideal gas 155 
Equilibrium constant 257 
Equipotential surface 50 
Equivalence, principle of 61 
Eutectic point 247 
Exothermic reaction 252 
Extension of solids 294-7 



Field, see Force field 
Finite motion 29 
Fluctuations 194 
Flux, electric 5 1 
Force 1 3 

centrifugal 83 

Coriolis 83 

dimensions of 17 

field 19,44-65 
electric 46-56 
gravitational 56-65 

inertia 81-85 

restoring 88 

resultant 77 

units of 17, 18 
Foucault pendulum 84 
Fractional distillation 244 
Frame of reference 1 

centre-of-mass 10 

inertial 2 

laboratory 34 
Free motion 1 
Freedom, degrees of 3,68 
Frequency of motion 86, 87 

angular 87 
Friction 94,316-17 

internal 341 



Gas 

constant 156 

ideal 153-60 

rarefied 357-60 
Gauss' theorem 51,59 
Glide plane 130 
Gravitation, Newton's law of 57 
Gravitational 

constant 57 

field 59-65 

interaction 56-65 

potential 58 
Gravity 60-62 

centre of 78 
Gyroscope 78-81 



Hardening 306,314 
Harmonic oscillations 86-93 
Heat 

content 1 70 

function 1 70 

of reaction 252-4 

of solution 230 

of transition 201 

see also Thermal; Thermodynamics 
Helium, liquid 200,218,225,361-6 
Henry's law 232 
Hooke'slaw 295 



Ideal gas 153-60 
Incompressible medium 350 
Indices of crystal planes 141-3 
Inertia 

forces 81-85 

law of 2 

moment of 69-7 1 
Inertial frame 2 
Infinite motion 29 
Internal energy 26-27 
Internal motion 26 
Inversion 

centre of 115 

point 184 
Ionisation potential 108 
Irreversibility of thermal processes 

188-9,192-4 
Isomorphism 235 
Isotopes 109-11 



Joule-Kelvin process 182-4 



370 



INDEX 



Kelvin scale of temperature 1 47 
Keplerian motion 63 
Kepler's 

first law 64 

second law 41 

third law 63 
Kinetic energy 

of particle 24 

of rigid body 68-70 

and temperature 145 
Kinetic processes 318 
Knudsen effect 360 



Laboratory frame of reference 3 4 

Laminar flow 352 

Latent heat 201 

Le Chatelier's principle 170 

Length, units of 15-16 

Lever rule 200 

Lines of force 48 

Liquid crystals 227-9 

Loschmidt's number 155 



Macroscopic properties 131,150 
Mass 

centre of 9, 78 

conservation of 8 

number 109 

of particle 6 

reduced 41 

of system 10 

units of 7,16-17,58 
Mass action, law of 257 
Maxwellian distribution 163 
Mean free path 328 
Mechanical properties of solids 294- 

317 
Meniscus 283 
Mixed crystals 234-6 
Mixing of liquids 232-4, 246-9 
Mobility 334 

Molecularity of reactions 270-2 
Molecules 111-14 

symmetry of 1 1 7-20 
Moment 

arm 37 

of couple 78 

of inertia 69-71 
Moments, law of 75 
Momentum 

angular, see Angular momentum 



of closed system 6 
conservation of 6 
of particle 6 
Monomolecular layer 280 



Nernst's theorem 177 
Newtonian mechanics 5 
Newton's 

first law 2 

law of drag 356 

law of gravitation 57 

second law 13 

third law 15 
Nucleus of atom 106 



One-dimensional motion 27 
Optical isomers 119 
Ordering of crystals 225-7 
Oscillations 86-104 

damped 93-96 

forced 96-102 

natural 89 

small 88-93 

torsional 93 
Osmosis 236-8 
Osmotic pressure 236-8 



Parametric resonance 102 
Particle 3 
Pascal's law 294 
Pendulum 

compound 91 

Foucault 84 

simple 90 

simple equivalent 92 
Periodic motion 29,86-104 
Perpetual motion 189 
Phase 

diagrams 198-200 

of matter 197 

of oscillation 87 

rule 251 

transitions 197-229 
of second kind 222-5 
Physical system of units 1 8 
Plane of symmetry 115 

glide 130 
Plastic deformations 306, 3 1 2- 1 5 
Poiseuille's formula 348 



INDEX 



371 



Poisson's 
adiabatic 181 
ratio 296 

Polymorphism 218 

Potential barrier 30 

Potential energy 22 
of electric charges 45 
of electric field 49 
of gravitational field 58 
in uniform field 23 

Potential well 30 

Power 26 

Precession 79 

Pressure 149-51 
osmotic 236-8 
partial 157 



Quantity of heat 168 

Radius vector 3 
Raoult's law 240 
Rarefied gases 357-60 
Reactive forces 7-8 
Relativity of motion 2 
Resonance 98-100 

parametric 102 
Restoring force 88 
Resultant force 77 
Reverse condensation 246 
Reversibility of motion 1 87 
Reynolds number 349-56 
Rigid body 66 

motion of 66-85 
Rigidity, modulus of 302 
Rotary -reflection axis 116 
Rotation of.rigid body 66-68, 72-76 



Saturated 

solution 230 

vapour 203-7 
Screw axis 130 
Sectorial velocity 41 
Self-diffusion 332 
Semipermeable membranes 236 
Separated flow 355 
Shear deformations 301-5 
SI system of units 18, 45, 50 
Similar motions 3 50 
Solid solutions 234-6 



Solidification of liquid mixtures 241-9 
Solubility 231 

product 263 
Solute 231 
Solutions 231-51 

dilute 23 1 

solid 234-6 

weak 231,237 
Solvent 23 1 
Space groups 1 29-3 1 
Specific heat 169-70 

of gases 171-4 

of solids and liquids 1 76-7 
Statistical weight 194 
Steady flow of gases and liquids 1 84-7 
Stereoisomerism 119 
Stokes' formula 351 
Sublimation 217 

Supercooling of vapours 207 , 2 8 9-9 1 
Superfluidity 361-6 
Superheating of liquids 207 ,290-1 
Superposition of fields 46 
Supersaturation of vapours 207, 

289-91 
Surface 

-active substances 280 

energy 276 

phenomena 276-93 

pressure 285 

tension 276-9,281-2 
Symmetry 115-43 

axis of 115,116,130 

centre of 115 

elements 115-16 

plane of 115,130 

translational 120 



Temperature 144-9 
Thermal 

conduction 319-34 

conductivity 320, 333-4, 358 

diffusion 336-8 

diffusivity 320 

equilibrium 144, 
193-4,254 

expansion 175-6 

motion 144 

processes 178-96 

resistance 324 

velocity 147 
Thermodynamic temperature 



151, 188-9 



145 



372 



INDEX 



Thermodynamics 

first law of 168 

second law of 189 
Time, units of 16 
Top 80-81 
Torque 38 
Torques, law of 75 
Torricelli's formula 187 
Torsion of rods 303-5 
Translation 

of crystal lattice 1 20 

of rigid body 66 
Transport phenomena 341 
Triple point 216 
Tube of force 52 
Turbulent flow 352-7 
Two-body problem 41 

Uniform compression 300 
Uniform deformation 297, 298-301 
Uniform extension 300 
Uniform field 

motion in 19-20 

potential energy in 23 
Unilateral compression 301 
Unit cell 120 
Units 15-18 

CGS system 18 

CGSE system 45, 50 

derived 15 

physical system 18 

SI system 18,45,50 



Van der Waals 

equation 211 

interaction 113-14 
Velocity 4 

addition rule 5 

angular, see Angular velocity 

cosmic 62, 64 

dimensions of 17 

of escape 64 

sectorial 41 

thermal 147 

units of 17 
Viscosity 341-66 

dynamic 342 

kinematic 342 



Weight 59 

Wetting of surfaces 283-5 

Work 20-26,166-7 

dimensions of 25-26 

units of 25-26 
Work-hardening 306, 3 14 



Yield point 308 
Young's modulus 



296 



Zero-point vibrations 148-9,173 



Collected Paj 
L,D. Landau 



rith an introduction by D. ter Haar 



The author of these papers occupies a 
position of international renown and is 
undoubtedly one of the world's most 
eminent theoretical physicists, He was 
awarded the Nobel Prize for Physics in 
1 962, The publication of his Collected 
Works is therefore an event of consider- 
able importance and this Volume contains 
a most comprehensive and significant 
collection of papers. Many of them have 
previously been very difficult to obtain 
and a large number of them have not been 
generally available in English. 

Landau's papers range over the whole of 
physics and cover such fields as Theory of 
liquid helium, Low temperature physics, 
Superconductivity, SoHd state physics, 
Plasma physics. Hydrodynamics, 
Astrophysics, Nuclear physics. Cosmic 
rays, Quantum mechanics and Quantum 
field theory. In an Appendix there is a list 
of papers by Landau which for various 
reasons were not included rn the Collected 
Papers. This book forms an essential 
addition to the shelves of every library 
concerned with physics and will also be a 
most valuable acquisition for many 
individual workers in the field. 
£10 net. 



Printed in Great Britain /5875/1 0/67 



^i 



tourse of Theoretical Physics 

.D. Landau and E.M, Lifshitz 
' tute of Physical Problems of the 
R Academy of Sciences 



The complete Course of Theoretical 
Physics* by Landau and Lifshitz, 
recognized as two of the world's out- 
standing physicists, is being published in 
full by Pergamon Press. It comprises nine 
volumes, covering all branches of the 
subject ; translations from the Russian are 
by leading scientists. 

Typical of many statements made by 

experts, reviewing the series, are the 

following: 

"The titles of the volumes in this series cover a 

vast range of topics, and there seems to be little in 

ph ysics on which the a uthors are not very weft 

in forme d." Nature 

". . . the whole set making up a Course of 

Theoretical Physics of remarkable completeness 

on both 'modern' and classical physics," Journal 

of Fluid Mechanics 

"The remarkable nine- volume Course of 

Theoretical Physics . . . the clearness and accuracy 

of the authors' treatment of theoretical physics is 

well maintained," Proceedings of the Physical 

Society 

"The monumental Course of Theoretical Physics. " 

Science Progress 

Of individual volumes, reviewers have 
written: 

Mechanics 

"The entire book is a masterpiece of scientific 
writing. There is not a superfluous sentence and 
the authors know exactly where they are going . . . 
it is certain that this volume will be able to hold its 
own amongst more conventional texts rn 
chassicaf mechanisms, as a scholarly and 
economic exposition of the subject. " Science 
Progress 

Quantum Mechanics 

(Non-relativistic Theory) 

"... throughout the five hundred large pages, the 
authors' discussion proceeds with the clarity and 
succinctness typical of the very best works on 
theoretical physics." Technology 

Fluid Mechanics 

"In the event, the book is one which will have to 
find its way onto the shelves of all those 
seriously interested in the subject. " Bulletin of 
the Institute of Physics 

The Classical Theory of Fields 

(Second Edition) 

"This is an excellent and readable volume, ft is a 
valuable and unique addition to the literature of 



theoretical physics." Science 

"The clarity of style, the conciseness of treatment, 

and the originality and variety of illustrative 

problems make this a book which can be highly 

recommended." Proceedings of the Physical 

Society 

Statistical Physics 

". , . stimulating reading, partly because of the 
clarity and compactness of some of the treatment: 
put forward, and partly by reason of contrasts with 
texts on statistical mechanics and statistical 
thermodynamics better known to English 
scientists ." NewScientist 

Theory of Elasticity 

"I shall be surprised if this book does not come to 
be regarded as a masterpiece. " Journalofthe 
Royal Institute of Physics 
".. .the book is well constructed, ably translated, 
and excellently produced." Journal of the Royal 
Aeronautical Society 

Electrodynamics of Continuous 
Media 

"Within the volume one finds everything expected 
of a textbook on classical electricity and 
magnetism, and a great deaf more It is quite 
certain that this book will remain unique and 
indispensable for many years to come. " Science 
Progress 

"The volume of electrodynamics conveys a sense 
of mastery of the subject matter on the part of the 
authors which is truly astonishing. " N atu re 

* Volume 4, Relativistic Quantum 

Mechanics, is almost completely written 
in Russian, and Pergamon expect to 
publish it during 1 968. Volume 9, 
Physical Kinetics, will be published at a 
future date when the manuscript in 
Russian is completed. 



Pergamon Press 

Headington Hill Hall, Oxford 

44-01 21st Street, Long Island City, 

New York 11101 

4 & 5 Fitzroy Square, London W1 

2 &3Teviol Place, Edinburgh 1 

6 Adelaide Street East, Toronto 

Rushcufters Bay, Sydney 

24 ruedes Ecoles, Paris 5e 

Vieweg 8i Sahn GmbH, Burgpiatz 1 . 

Braunschweig 



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